LIBRARV OF THK University of California. Received (£//^yr . /^ Accession No. £ O J^ J/l . Class No. ARITHMETIC FOE SCHOOLS. ARITHMETIC FOE SCHOOLS By BARNARD SMITH, M.A. RRCTOR OF GLASTON, RUTLAND, LATE FELLOW AND SENIOR BURSAR OF ST PETER'S COLLEGE. CAMBRIDGE. J4EW ECJITION. ^K^^ Of THM ^ Hontron: MACMILLAN AND CO. i8;3 [All Rights reserved.] PRINTED BY C. J. CLAY, M. A. AT THE UNIVERSITY PRESS. ADVERTISEMENT. This Edition, wliicli is printed from the stereotype plates, has been carefully revised. If any gentleman, wlio may happen to meet with errors, will kindly communicate them to the Publishers, he will confer a great obligation on the Author. St. Peter's College, Cambridge. PREFACE TO THE EDITION OF 1865 OF BARNARD SMITH'S ARITHMETIC FOR SCHOOLS. A. An Act of Parliament having been passed last Session, legalizing the use of the Metric System of Weights and Measures, the Author has deemed it advisable to publish a Companion to his Arithmetic for Schools, now in the Press, containing, besides other new matter, the Metric System, and its application, the Money Tables of the Principal States of Europe, America, and India, and their application. Mental Arithmetic, Logarithms, the application of Arith- metic to Geometry. Gl.vston Eectoet, January 24, 1865. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/arithmeticforschOOsmitrich CONTENTS. PAGB Definitions, Notation, and Numeration i . . . . i Addition ........... 4 Simple Addition ....;;.... 5 Subtraction ...;....•• 8 Simple Subtraction . . ; . . . • • . lO. Multiplication . . ; * H Simple Multiplication . ; I2 Division ........... 17 Simple Division ' . . . ib. Greatest Common Measure ....... 24 Least Common Multiple . . . . . . * * '2^ Miscellaneaus Questions and Examjples on Arts, i — 57 . . 31 Fractions 34 Vulgar Fractions ......... 35 Addition of Vulgar Fractions . . ; i i . . 45 Subtraction of Vulgar Fractions . . . i i i 48 Multiplication of Vulgar Fractions ...... 5® Division of Vulgar Fractions ..*;;.. 52 Miscellaneous Examples, ivorlced out ...... 55 Miscellaneous Questions and Examples on Arts. 58 — 79 . . 58 Decimals .... i ...... 62 Addition of Decimals 65 Subtraction of Decimals ...... i * 67 Multiplication of Decimals ......* 68 Division of Decimals ......... 6g Vulgar Fractions expressed as Decimals . . ; . 75 Circulating Decimals . . ib. Miscellaneous Questions and Examples on Arts. 80 — 99 . . 81 Concrete Numbers (Tables). Money . . . . . .85 Measures of Weight ........ 87 " Length 89 " Surface 90 " Solidity , . 91 Vlll CONTENTS. PAQB Measures of Capacity , 92 " Number ......... 93 " Time ih. Reduction ........... 96 Compound Addition ........ 103 " Subtraction . . . . . . . .108 " Multiplication . . . . , . . iii *' Division 114 Miscellaneous Examples ivorhed out . . . . . . 121 Decimal Coinage . . . . . . . . ,125 Miscellaneous Questions and Examples on Arts, ico — 131 . . 129 Reduction of Fractions . . . . . . . • I37 " Decimals ........ 147 Practice 156 Square and Cubic Measure. Cross Multiplication or Duodecimals 161 Miscellaneous Examples, worJced out . . . . . . 175 Miscellaneous Questions and Examples on Aris.x^2 — 141 . . 183 Rule of Three 192 Double Rule of Three . . . 212 Interest ........... 227 Simple Interest .......... t&. Compound Interest . . . . . , . . . 233 Present "Worth and Discount . . . . . . .236 Present Worth . . 237 Discount ........... »6. Stocks 240 Profit and Loss .......... 247 Division into Proportional Parts . ., . . . . 251 Fellowship or Partnership . . ... . . . •253 Simple Fellowship . . ... . . . . ib. Compound Fellowship ib. Equation of Payments . . ' . . . . . . 254 Applications of the Term Per Cent. . . . . . • ^57 Exchange 264 Square Root 266 Cube Root 274 Miscellaneous Questions and Examples . .... 283 Appendix. Miscellaneous Papers ........ 299 ANSWERS 315 ARITHMETIC, DEFINITIONS, NOTATION, AND NUMERATION. Article 1. By a Unit is meant a single object or thing, considered as one and undivided. 2. Number is the name by which we signify how many objects or things are considered, whether one or more. When, for instance, we speak of one horse, two apples, three yards, or four hours, the number of the things referred to will be one, two, three, or four, according to the case ; and so one, two, three, four, and the rest, are called numbers. 3. NujiBERS are considered either as Abstract or Concrete. Abstract numbers are those which have no reference to any particular kind of unit ; thus, five, as an abstract number, signifies five units only, without any regard to particular objects. J Concrete numbers are those which have reference to some particular ^ kind of unit ; thus, when we speak of five hours, six yards, seven horses, the numbers five, six, seven, are said to be concrete numbers, having reference to the particular units one hour, one yard, one horse, respec- tively. 4. Arithmetic is the science of Numbers. 5. All numbers in common Arithmetic are expressed by means of the figure 0, commonly called zero or a cypher, which has no value in itself, and nine significant figures, 1, 2, 3, 4, 5, 6. 7, 8, 9, which denote respectively the numbers one, two, three, four, five, six, seven, eight, nine. These ten figures are sometimes called Digits ; but this name is often improperly limited to the nine significant figures above mentioned, which are then called the nine digits. The number one, which is represented by the figure 1, is called UNITY. G. "\rhcn any of these figures stands by itself, it expresses its simple or intrinsic value \ thus, 9 expresses nine abstract unit?, or nine particular 2 ARITHMETIC. things : but when it is followed by another figure, it then expresses ten times its simple value ; thus, 94 expresses ten times nine units, together with four units more : when it is followed by two figures, it then expresses one hundred times its simple value ; thus, 943 expresses one hundred times nine units, together with ten times four units, and also three units more: and so on by a tenfold increase for each additional figure that follows it. The value, which thus belongs to a figure in consequence of its posi- tion or place, is called its local value. Therefore all numbers have a simple or intrinsic value, and also a local value. 7. It appears then, that in common Arithmetic we proceed towards the left from units to tens of units ; from tens of units to tens of tens of units, or hundreds of units ; from hundreds of units to tens of hundreds of units, or thousands of units ; from thousands of units to tens of thousands of units ; from tens of thousands of units to tens of tens of thousands of units, that is, to hundreds of tliousands of units ; thence to tens of hundreds of thousands of units, or millions of units ; thence to tens of millions of units, hundreds of millions of units, &c., till we come to millions of millions of units, which are called billions of units, and so on to trillions, quadrillions, &c. Thus, 10 represents one ten of units, together with no units ; or, as it is briefly read, ten. 11 represents one ten of units, together with one unit ; or, as it is briefly read, eleven. Similarly 12, 13, 14, 15, 16, 17, 18, 19, respectively represent one ten of units together with two, three, four, five, six, seven, eight, nine units ; they are respectively read twelve, thir- teen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. The next ten numbers are expressed by 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, which respectively represent two tens of units together with no, one, two, three, four, five, six, seven, eight, nine units ; they are briefly read twenty, twenty-one, twenty-two, twenty-three, twenty-four, twenty- five, twenty-six, twenty-seven, twenty-eight, twenty-nine. The next ten numbers are expressed by 80, 31, 82, 33, 34, 35, ^Q, 37, 38, 89, which are respectively read thirty, thirty-one, thirty-two, thirty- three, thirty-four, thirty-five, thirty-six, thirty-seven, thirty-eight, thirty- nine : we thus arrive at 40 (forty), 50 (fifty), 60 (sixty), 70 (seventy), 80 (eighty), 90 (ninety). 99 is the largest number which can be expressed by two figures, since it represents nine tens of units together with nine units ; tlie next number DEFINITIONS, NOTATION, AND NUMERATION. J to this is 100,. which represents ten tens of units, or one hundred of units, together with no tens of units, together with no units ; or, as it is briefly read, one hundred. By pursuing tlie same system in higher numbers the figure occupying the fourth place from the right hand will represent so many tens of hun- dreds of units, or thousands of units ; the figure in the fifth place will represent so many tens of thousands of units ; and so on. 205 represents two hundreds of units, together with no tens of units, together with five units ; or, as it is briefly read, two hundred and five. 5473 represents five thousands of units, together with four hundreds of units, together with seven tens of units, together with three units ; or, as it is briefly read, five thousand, fonr hundred and seventy-three. 70-10730 represents seven millions of units, together with no hundreds of thousands of units, together with four tens of thousands of units, toge- ther with no thousands of units, together with seven hundreds of units, together with three tens of units, together with no units ; or, as it is briefly read, seven millions, forty thousand, seven hundred and thirty. 107834265 represents one hundred of millions of units, together with no tens of millions of units, together with seven millions of units, together with eight hundreds of thousands of units, together with three tens of thousands of units, together with four thousands of units, together with two hundreds of units, together with six tens of units, together with five units ; or, as it is briefly read, one Inmdred and seven millions, eight liimdred cmd thii-ty-four thousand, two hundred and sixty-five. 8. Notation is the art of expressing any number by figures which is already given in words. Numeration is the converse of Notation, being the art of expressing any number in words which is already given in figures. 9. The method above explained of denoting numbers by means of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and combinations of them, was brought into Europe by the Arabs, and it is therefore often called the Arabic Notation. It was derived by the Arabs from the Hindoos. This method of notation is now in common use, not only in this country, but throughout Europe. Ex. 1. Ejcercises in Notation and Numeration. Express the following numbers in figures : (1) Sixty- three; eighty -one ; ninety- nine; forty; thirteen. 1—3 4 ARITHMETIC. (2) Two hundred; three hundred and three; seven hundred and sixty-four ; eight hundred and eighty-eight. (3) Four thousand ; one thousand, four hundred and seventy-one ; eix thousand, nine hundred and thirty j nine thousand and nine. (4) Twenty-seven thousand, five hundred and four; thirty-three thousand; nine thousand and sixteen. (5) One hundred thousand ; six hundred and seventy-six thousand and fifty; two hundred and two thousand, five hundred and ninety- three. (6) Seven milUons, three thousand; eleven millions, one hundred and eight thousand, one hundred and six ; fifty-four millions, fifty-four thousand and eighty-eight ; six hundred and thirteen millions, twenty thousand, three hundred and three. (7) Two billions ; nine billions, three hundred thousand and twenty- one; ninety-four billions, ninety millions, ninety-four thousand, nine hundred and four. Write down in words at full length the following numbers : (1) 43; GO; 88; 97; 59; 12; 21; 19. (2) 256; 401; 500; 999; S65 ; 578; 837. (3) 2000; 1724; 8003; 7584; 1075; 4541. (4) 87003; 47049; 63090; 80008; 841823. (5) 6850406; 8080808; 7849630; 418254. (6) 10000001 ; 20220022 ; 92568987 ; 80180070, (7) 2560530200; 800309560; 9788413208. (8) 7070000423 ; 987654321 ; 5707068080. (9) 100198700010090 ; 48726870634108264. ADDITIOK 10. Addition is the method of finding a number, which is equal to two or more numbers taken together* The number foimd by adding two or more numbers together is called the SUM or ajiount of the several numbers so added. 11. There are two kinds of Addition, Simple and Compound. It is Simple Addition, when the numbers to be taken together are all abstract numbers ; or when they are all concrete numbers of the same de- nomination, as all pence, all days, all pints. It is Compound Addition, when the numbers to bo taken together are ADDITION. 6 concrete numbers of the same kind, but of different denominations of that kind ; as pounds, shillings, and pence ; or years, months, and days ; or gallons, quarts, and pints. 12. The sign + , plus, placed between two or more numbers, signifies that the numbers are to be added together : thus 2 + 6 + 7 signifies that 2, 5, and 7 are to be added together, and denotes their sum. The sign = , equal, placed between two numbers, signifies that the numbers are equal to one another. The sign , viNcuLU3r, placed over numbers, and the sign ( ) or { [, called a bracket, enclosing numbers within it, are used to denote that all numbers under the vinculum, or within the bracket, are equally affected by all numbers not under the vinculum or within the bracket : thus 2 + 3 or (2 + 8) or {2 + 3}, each signify, that whatsoever is outside the vinculum or bracket which affects 2 in any way, must also affect 3 in the same v/ay, and conversely. The sign .*. signifies ' therefore.' SIMPLE ADDITION. 18. Rule. Write down the given numbers under each other, so that units may come under units, tens under tens, hundreds under hundreds, and so on ; then draw a straight line under the lowest line. Find the sum of the column of units ; if it be under ten, write it down under the column of units, below the line just drawn ; if it exceed ten, then write down the last figure of the sum under the column of units, and carry to the next column the remaining figure or figures ; treat each succeeding column in the same way, and wi-ite dowTi the full sum of the extreme left-hand column. The entire sum so marked down will be the sum or amount of the separate numbers. 14. Add together 5469, 743, and 27. Proceeding by the Rule given above, we obtain 6469 743 27 G239 2Vie reason for the Rule will appear from the following considerations. When we take the sum of 7 units and 3 units and 9 units, we get 10 (5 ARITHMETIC. units and 9 units, or 19 units ; we therefore place the 9 units under the column of units and carry on the 1 ten units to the next column, viz. the column of tens. Now the sum of 1 ten, 2 tens, 4 tens, and G tens, is 10 tens and 3 tens, or 13 tens ; we therefore place the 3 tens under the column of tens and carry on the 1 hundred units to the next column, viz. the column of hundreds. Again, the sum of 1 hundred, 7 hundreds, and 4 hundreds, is 10 hun- dreds and 2 hundreds, or 12 hundreds ; we therefore place the 2 hundreds under the column of hundreds, and carry on the 1 thousand units to the next column, viz. the column of thousands. Again, the sum of 1 thousand and 5 thousands, is G thousands ; wo therefore place the 6 under the column of thousands, and the entire sum is 6239. 15. The above example might have been worked thus, putting dowTi at full length the local value of all the figures. Thus 5469 = 5000 + 400+60 + 9 + 743= +700 + 40 + 3 + 27= +20 + 7. Now adding the columns, we get the sum -=5000 + 1100 + 120 + 19 = 5000 + 1000 + 100 + 100 + 20 + 10 + 9, (since 1100 = 1000 + 100, 120 = 100 + 20, and 19 = 10 + 9 ) = 6000 + 200 + 30 + 9, (collecting the thousands together, the hundreds together, and so on) = 6239. Note. The truth of all results in Addition may be proved by adding the columns first upwards as in the above example, and then adding them downwards ; if the results be the same, the operation in each case will in all probability have been performed correctly. Ex. II. Examples in Simple Addition. (1) 12 35 (2) 57 87 (3) 234 567 (4) 654 321 5G 65 753 804 80 m 43 345 ^09 SIMPLE ADDITION". (5) 494 f6) 1721 (7) 750 (8) 4789 587 3333 36 2346 656 5046 18t3 3857 336 2754 561 3100 6005 (9) 9102 (10) 84670 (n) 1790621 (12) 256783 479 5437 206803 21003 8776 29 353 5734 901 21904 9003766 40036 21 100001 428578 (13) G27432 (14; 892764 (15) 1807353 (16) 117064 648201 93687 298743 92073 678641 9482 5987 827569 548200 100 760003 351 8G8759 . 152346 247 777777 345678 11 50705 65656 (17) Add together 7384, 326, 6780, and 57 ; also 6740, 9745, 5769, 8031, 6543, 2002, and 9999 ; also 89, 4500, 423, 2024, 5408, 60546, and 9401. (18) Add together 83746, 2478, 692577, 456, and 7; also 935473, 262, 18897, 598453, 25, 3734, 724008, and 649768. (19) Find the sum of 4738685, 237869513, 148794343978, 865, 4647, and 250; also of 68539582, 78602045, 370489000, 7055591234, 276, 9123456789, and 5000: also of 888929944, 73600, 27978462, 333, 5875396006, 4827582, 486684836, 80032148379, 12345, 1112858673, and 63800000835. (20) Add together one thousand, four hundred and eighty-three; seven hundred and ninety-six ; thirty-nine ; forty thousand, seven hun- dred and forty-four ; five thousand, eight hundred and sixty ; fifty • thousand and seven. (21) Add together the following numbers: fifteen thousand, seven hundred and ninety-six ; four hundred and nine ; two hundred and thirty- four thousand and fifty ; four millions, three thousand and seventy-six; forty thousand and thirty-six ; ten thousand, nine hundred and one. (22) Add together the following numbers : twenty-two millions, six hundred thousand, five hundred and three ; five hundred and sixty-three 8 ARITHMETia millions, seventy-six thousand and thirty-four ; one hundred and eleven millions, six hundred and fifty thousand and fifty ; three hundred and twenty-six millions, seven thousand, nine hundred and ninety-one ; one thousand seven hundred and ten millions, one thousand seven hundred and ten ; one billion, three hundred thousand and five. SUBTKACTIOK. 16. Subtraction is the method of finding v/hat number remains when a smaller number is taken from a greater number. The number found by subtracting the smaller of two numbers from the greater is called the Remainder. 17. There are two kinds of Subtraction, Simple and Compound, which differ from each other in precisely the same way, in which Simple and Compound Addition differ from each other, 18. The sign—, minus, placed between two numbers, signifies that the second number is to be subtracted from the first number. SIMPLE SUBTRACTION. ID. Rule. Place the less number under the greater number, so that units may come under units, tens under tens, hundreds under hundreds, and so on ; then draw a straight line under the lower line. Take, if possible, the number of units in each figure of the lower line from the number of units in each figure of the upper line which stands immediately over it, and put the remainder below the line just drawn, units under units, tens under tens, and so on : but if the units in any figure in the lower line exceed the number of units in the figure above it, add ten to the upper figure, and then take the number of units in the lower figure from the number in the upper figure thus increased ; put the remainder down as before, and then carry one to the next figure of the lower line. The entire difference or remainder, so marked down, will be the difference or remainder of the given numbers. 20. Ex. Subtract 4938 from 5123. Proceeding by the Rule given above, we obtain 6123 4938 185 SO that the remainder is one hundred and eighty-five (185). SUBTRACTION, 9 The reason for the Rule will appear from the following considerations. We cannot take 8 units from G units, we therefore add 10 units to the 8 units, which are thus increased to 13 units ; and taking 8 units from 13 units we have 5 units left ; we therefore place 5 under the column of units: but having added 1 fen units to the upper number, we must add the same number of units (1 ten units) to the lower number, so that the difference between the two numbers may not be altered ; and adding 1 ten units to the 8 ten units in the lower number, we obtain 4 tens or 40 instead of 8 tens or 30. Again, we cannot take 4 tens from 2 tens ; we therefore add 10 tens 01* 1 hundred to the 2 tens, which thus become 12 tens or 120 ; and then taking 4 tens or 40 from 12 tens or 120, we have 8 tens or 80 remaining ; we therefore place 8 under the column of tens : but having added 1 hun- dred to the upper number, we must add 1 hundred to the lower number for the reason given above ; and adding 1 hundred to the 9 hundreds in the lower number, we obtain 10 hundreds or 1000 instead of 900. Again, M-e cannot take 10 hundreds from 1 hundred, and we therefore add 10 hundreds or 1 thousand to the 1 hundred, which thus becomes 11 hundreds or 1100 : and taking 10 hundreds or 1000 from 11 hundreds or IIDO, we have 1 hundred or 100 left; we therefore place 1 under the column of hundreds : but having added 10 hundreds or 1 thousand to the upper number, we must add 1 thousand to the lower number for the reason given above ; and adding 1 thousand to the 4 thousands in the lower number, we obtain 5 thousands or 5000 ; 5000 taken from 5000 leaves ; therefore the whole difference or remainder is 185. 21. The above Example might have been worked thus, putting down at full length the local values of the figures : 6123= 5000 -fl00+ 20 +3 4000 + 1000 + 1004- 20 +8 = 4000 + 1000 + 100 + 10 + 10 + 3 = 4000 + 1000 + 110 + 13 (collecting the first 10 with the 100, and the second 10 with the 3,) 4938 = 4000 -4- 900 + 30 + 8. Therefore subtracting the columns, thousands from thousands, &c. we get the remainder or difference = 100 + 80 + 5 = 185. 10 ARITHMETIC. Note. The truth of all results in Subtraction may be proved by adding the less number to the difference or remainder ; if this sum equals the larger number, the result obtained by subtraction may be presumed to be correct. I :x. III. Examples in Simple Subtraction. (1) G63 580 83 (2) 976 631 (3) 704 483 (4) 806 720 (5) 4236 8089 (6) 80502 38f>72 (7) 46095 28736 (8) 555555 123456 (0) 1000000 (10) 100101 400357261 (11) 99988877 89437182 15790293 (12) Find the difference between 6543756 and 412848 ; 7863927 and 826957; 803233334 and 192001222. (13) How much greater is 164326289 than 48476798 ? 10000001000 than 7077070077? 7559030640021 than 6990040005679? (14) Take two thousand and nine, from ten thousand and ninety- six ; three thousand and eight, from seven thousand, nine hundred and forty-four. (15) Required the difference between four and four millions; also between one hundred millions and three hundred thousand. (16) Subtract five hundred and eighty-four thousand and seventy-six, from fifteen millions, one hundred thousand and three. 22. The following method of expressing numbers was used by the Romans, and it is still in occasional, though not in common use, among ourselves. They represented the number one by the character I ; five by V ; ten by X ; fifty by L ; one hundred by C ; five hundred by D or I j ; one thousand by M or CIq. All other numbers were formed by a combinallon of the above charac- ters, subject to the following Rules : First ; When a character was followed by one of equal or less value, the whole expression denoted the sum of the values of the single charac- ters ; for instance, II stood for 2 ; III for 3 ; VI for 6 ; VIII for 8 ; LV for 55 ; LXXVII for 77; CCXI for 211. Secondly ; When a character was preceded by one of less value, the MULTIPLICATION. 1 1 whole expression denoted the difference of the vaUies of the single charac- ters ; for instance, IV stood for 5 — 1, or 4; IX for 10 — 1, or 9; XIX for 10 + 10-1, or 19; XL for 50-10, or 40 ; XC for 100-10, or 90. Thirdly ; Every 3 annexed to 1 3 increased the value of the latter tenfold ; for instance, 133 stood for 5000; I333 for 50000 ; and so forth. And every C prefixed and 3 annexed to CI3 increased the value of the latter tenfold ; for instance, CCI33 stood for 10000 ; CCCI333 for 100000 ; and so forth. Fourthly ; A line drawn over a character or characters increased the value of the latter a thousandfold ; for instance, V stood for 5000 ; C for 100000 ; IX for 9000 ; and so forth. It follows then that either XXXXVI or XLVI will represent 46 : and that either M.DCCC.LIV, or CI3.I3CCCLIV, or T.DCCCLIIII will represent 1854. E3?. IV. (1) Express in Roman characters, thirty; forty-eight; fifty-nine; 222; 6000; 1843. (2) Express in words, and also in A rabic figures, the values of XXIII ; LXIX; CCXVIII; VI; CLDCIII; MMC. MULTIPLICATIOK 23. Multiplication is a short method of finding the sum of anj*^ given number repeated as often as there are units in another given number; thus, when 3 is multiplied by 4, the number produced by the multiplica- tion is the sum of 3 repeated 4 times, which sum is equal to 3 + 3 + 3 + 3 or 12. The nuinber to be repeated or added to itself, is called the Multipli- cand, The number which shews how often the multiplicand is to be repeated or added to itself, is called the Multiplieii. The number found by multiplication is called the Product. The multiplicand and multiplier are sometimes called * Factors,' be- cause they are factors or makers of the product, 24. Multiplication is of two kinds. Simple and Compound. It is termed Simple Multiplication, when the multiplicand is either an abstract number, or a concrete number of one denomination. It is termed Compound Multiplication, when the multiplicand contains numbers of more tjian oiie denomination, but all of the s^me kind. 12 ARITHMETIC. 25. The sign x, placed between two numbers^ signifies that the numbers are to be multiplied together. 26. The following Table ought to be learned correctly t 1 2 3 4 5 6 7 8 9 10 11 12 2 4 G 8 10 12 14 IG 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 8 12 IG ^0 24 28 32 35 40 44 48 5 10 15 20 25 30 '35~ 40 IL 50 55 60 6 12 18 24 30 36 42 48 54 "eo" 66 72 7 14 21 Hf 35 42 49 5G 63 70 77 84 8 16 24 32 40 48 5G 64 72 80 88 96 9 18 27 3G 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 SO 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 In the above Table^ the second line from the top shews the product of each of the numbers, 1, 2, S, 4, &c. 11, 12, in the first line, when multi- jjlied by 2 ; the several products being placed under the respective num- bers of the line above, from the multiplication of which they arise : the third line shews the several products, when the figures in the first line are respectively multiplied by 8 : and so on. N'ote. One of the factors, namely the multiplier, must necessarily be an 'abstract number*; since it would be absurd to speak of 6 shillings multiplied by 4 shillings. We can multiply 6 shillings by 4, i. e. we can find how many shillings there are in four times six shillings ; but there is no meaning in 6 shillings multiplied by 4 shillings. SIMPLE MULTIPLICATION. 27. Rule. Place the multiplier under the multiplicand, units under units, tens under tens, and so on. Multiply eacli figure of the multipli- cand, beginning with the units, by the figure in the units' place of the multiplier (by means of the table given for Multiplication) ; set down and carry as in Addition. Then multiply each figure of the multiplicand, SIMPLE MULTIPLICATIONS. 13 beginning with the units, by the figure in the tens' place of the multiplier, placing the first figure so obtained under the tens of the line above, the next figure under the hundreds, and so on. Proceed in the same way with each succeeding figure of the multiplier. Then add up all the results thus obtained, by the rule of Simple Addition. Note. If the multiplier does not exceed 12, the multiplication can be effected easily in one line, by means of the Table given above- 28. Ex. Multiply 7654 by 897. Proceeding by the Rule given above, we obtain 7654 397 58.578 ' 68886 22962 3088638 The reason for the Rule will appear from the following considerations. When 7654 is to be multiplied by 7, we first take 4 seven times, which Ijy the Table gives 28, i.e. 8 units and 2 tens ; we therefore place down 8 in the units' place and carry on the 2 tens :^ again, 5 tens taken 7 times give 35 tens, to which add 2 tens, and we obtain 37 tens, or 7 tens and 3 hundreds ; we put down 7 in the tens' place, and carry on 3 hun- dreds ; again, 6 hundreds taken 7 times give 42 hundreds, to which add 3 hundreds, and we obtain 45 hundreds, or 4 thousands and 5 hundreds ; we put down 5 in the hundreds' place, and carry on the 4 thousands : again, 7 thousands taken 7 times give 49 thousands, to which we add the 4 thousands, thus obtaining 53 thousands, which we write do\^'n. Next, when we multiply 7654 by the 9, we in fact multiply it by 90 ; and 4 units taken 90 times give 360 units, or 3 hundreds, 6 tens, and units : therefore, omitting the cypher, we place the G under the tens' place, and carry on the 3 to the next figure, and proceed with the operation as in the line above, "When we multiply 7654 by the 3, we in fiict multiply by 300 ; and 4 multiplied by 300 gives 1200, or 1 thousand, 2 hundreds, tens, and units ; therefore, omitting the cyphers, we place the first figure 2 under the hundreds' place, and proceed as before. Then adding up the three lines of figures which we have just obtained, we obtain the product of 7654 by 397. 14 ARITHMETIC. 29. The above Example might have been worked thus, putting down at full length the local values of the figures ; 7654= 7x1000+ 6x100+ 5x10+ 4 397= 3x100+ 9x10+ 7 49x1000 + 42x100 + 35x10+28 63x10000+ 54x1000 + 45x100 + 36x10 21 X 100000+18 x 10000 + 15 x 1000 + 12 x 100 21 X 100000 + 81 X 10000 + 118 x 1000 + 99 x 100 + 71 x 10 + 28 which = 20x100000+ 1x100000 + 8x100000 + 1x10000 + 1x100000 + 1x10000+ 8x1000 + 9x1000+ 9x100 + 7x100 + 1x10 + 2x10 + 8 2000000 + 10 X 100000 + 2 X 10000+17 x 1000 + 16 x 100 + 3 x 10 + 8 = 2000000 + 1000000 + 2x10000 + 10x1000 + 7x1000 + 10x100 + 6x100+3x10+3 = 3000000 + 2 X 10000 + 1 x lOUOO + 7 x 1000 + 1 x 1000 + 6 X 100 + 3 x 10 + 8 = 3000000 + 3 X 10000 + 8 x 1000 + 600 + 30 + 8 = 3000000 + 30000 + 8000 + 600 + 30 + 8 = 3038638 80. If the multiplier or multiplicand, or both, end with cyphers, we may omit them in the working ; taldiig care to affix to the product as many cyphers as we have omitted from the end of the multiplier or multiplicand, or both. Thus, if 2G3 be multiplied by G200, and 570 bo multiplied by 3200, we have 263 670 " 6200 ' 3200 526 114 1578 171 1630600 1824C00 The reason is clear : for in the first case, when we multiply by the 2, in fact we multiply by 200 ; and 3 multiplied by 200 gives 600 : in the second case, the 7 multiplied by the 2 is the same as 70 multiplied by 200 ; and 70 multiplied by 200 gives 14000. 31. If the Multiplier contain any cypher in any other place, then, in multiplying by the different figures of the multiplier we may pass over the cypher ; taking care, however, when we multiply by the next figure, to place the first figure arising from that multiplication under the SIMPLE MULTIPLICATION. 15 third figure of the line above instead of the second figure. The reason of this is clear : for, if we were multiplying by 206, when we muUiply by the G we take the multiplicand 6 times, when we multiply by the 2 we really take the multiplicand, not 20 times, but 200 times. 82. When two numbers are to be multiplied together, it is a matter of indifference, so far as the product is concerned, which of them be taken as the multiplicand or multiplier ; in other words, the product of the first multiplied by the second, will be the same as the product of the second multiplied by the first. Thus, 2x4 = 2 + 2 + 2 + 2-8, 4x2=4+4 =8; therefore the results are the same, that is, 2x4 = 4x2. That the product of one number multiplied by another, will be equal to the product of the latter multiplied by the former, may perhaps appear more clearly from the following mode of shewing this equality in the case of the numbers 3 and 5. 3=1+1+1; .-. 3 x5 = (l + l + !) + (! + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) =1+1+1 +1+1+1 +1+1+1 +1+1+1 +1+1+1 15. Now, if we regard the ones from left to right, there are 3 ones taken 5 times ; if we regard them taken from top to bottom, we have 5 ones re- peated 3 times ; and the number of ones in each case is the same ; i. e. 3x5 = 5x3: and so in the case of any two other numbers multiplied together, 33. The truth of all results in Multiplication may be proved by using the multiplicand as multiplier, and the multiplier as multiplicand: if the product thus obtained be the same as the product found at first, the results are in all probability true. 34. We have hitherto confined our attention to products formed by the multiplication of two factors only. Products may however arise from the multiplication of three or more factors; this is termed Continued Multiplication : thus 2x3x4 denotes the continued multiplication of the fiictors 2, 3, and 4 ; and means that 2 is to be first multiplied by 3, 16 ARITHMETIC. and the product thus obtained to be then multiplied by 4. The result of such a process would be 24, which is therefore the continued product of 2; 3, and 4 : we may express it thus, 2 x 3 x 4=24. Ex. V. Examples in Simple Multiplication. (1) 534 4 2136 (2) 673 3 (3) 2867 5 (4) 7492 6 (5) 2057 7 (6) 57409 8 (7; 2745638 9 (8) 5763 11 63393 (9) 35976 11 (10) 91525 12 (11) 257 53 771 1285 13621 (12) 96843 17 (13) 87298 46 (14) 16097 59 (15) 296897 83 (16) 69284 90 6235560 (17) 840607 8C (18) 175 ) 189 (21) 25607 5004 (19) 6298 769 (22) 78847 8803 (20) 5423 603 16269 32588 3270069 (23) Find the product of 234578 by 18, by 29, and also by 53 ; of 924846 by 67, by 95, and' also by 430; 2846067 by 206, by 1008, and also by 907 ; 8409681 by 21711, by 7009, by 8435, and also by 7980. (24) Find the product of 1754 and 9306; of 47506 and 4500; of 149570 and 15790 ; of 554768 and 39314 ; of 815085 and 20048 ; of 123456789 and 987654321 ; and of 57208492692 and 700809050321. (25) Multiply 9487352 by 4731246 ; 4342760 by 599999 ; 17376872 by 7399078 ; 38015732 by 400700065 ; 574585614865 by 2837154309. (26) Multiply six hundred and fifty thousand and ninety, by three thousand and eight; also seventy-six millions, eight thousand, seven hundred and sixty-five, by nine millions, nine thousand and nine. DIVISION. 17 (27) Find the continued product of 12, 17, and 19 ; of 3781, 3782, and 3783 ; and of 6oGo, 6786, and 9898. (28) Multiply 20470 by 1030, and 2958 by 476, explaining the reason of each step in the process. BIVISIOK 35. Division is the method of finding how often one number, called the Divisor, is contained in another number, called the Dividend. The result is called the Quotient. 36. Division is of two kinds. Simple and Compound. It is called Simple Division, when the dividend and divisor are, both of them, either abstract numbers, or concrete numbers of one and the same denomination. It is called Compound Division, when the dividend, or when both divisor and dividend contain numbers of different denominations, but of one and the same kind. 37. The sign -f-, placed between two numbers, signifies that the first is to be divided by the second. 38. In Division, if the dividend be a concrete number, the divisor may be either a concrete number or an abstract number, and the quotient will be an abstract number or a concrete number, according as the divisor is concrete or abstract. For instance, 5 shillings taken 6 times give 30 shillings, therefore 30 shillings divided by 5 shillings give the abstract number 6 as quotient ; and 80 shillings divided by 6 give the concrete number 5 shillings as quotient. SIMPLE DIYISION. 39. Rule. Place the divisor and dividend thus : divisor) dividend (quotient. Take off from the left-hand of the dividend the least number of figures which make a number not less than the divisor ; then find by the Multi- plication Table, how^ often the first figure on the left-hand side of the divisor is contained in the first figure, or the first two figures, on the left- hand side of the dividend, and place the figure which denotes this number of times in the quotient : multiply the divisor by this figure, and bring down the product, and subtract ;t from tlie number which was taken off 2 is AKITHMETIC. at the left of the dividend : then bring down the next figure of the divi- dend, and jjlace it to the right of the remainder, and proceed as before; if the divisor be greater than this remainder, affix a cypher to the quotient, and bring down the next figure from the dividend to the right of the remainder, and proceed as before. Carry on tliis operation till all the figures of the dividend have been thus brought down, and the quotient, if there be no remainder, will be thus determined, or if there be a re- mainder, the quotient and the remainder will be thus determined. Note 1. If any product be greater than the number which stands above it, the last figure in the quotient must be changed for one of smaller value ; but if any remainder be greater than the divisor, or equal to it, the last figure of the quotient must be changed for a greater. Note 2. If the divisor does not exceed 12, the division can easily b© effected in one line, by means of the Multiplication Table. 40. Ex. Divide 23882G8 by 6758. Proceeding by the Rule given above, we obtain 6758) 2338268 (346 20274 31086 27032 40548 40548 Therefore the quotient is 346. The reason for the Rule will appear from the following considerations. The divisor represents six thousand, seven hundred and fifty-eight ', the first five figures on the left-hand side of the dividend represent two millions, three hundred and thirty-eight thousand, and two hundred. Now the divisor is contained in this 300 times; and 6758 x 300 = 2027400, Or omitting the two cyphers at the end for convenience in workmg, we properly place the 4 under the 2 in the line above ; we subtract the pro- duct thus found, and we obtain a remainder of 8108, which represents three hundred and ten thousand, and eight hundred. . Bring down the 6 by the Rule; this 6 denotes 6 tens or 60, but the cypher is omitted for the reason above stated : the number now represents three hundred and ten thousand, eight hundred and sixty : 6758 is contained 40 times in this, and 6758x40 = 270320; we omit the cypher at the end as before, and subtract the 27032 from the- 31086 ; and after subtraction the remainder is 4054, which represents forty thousand, five hundred and forty, Bring SIMPLE DIVISION. 19 down the 8 by the Rule, and the number now represents forty thousand, five hundred and forty-eight : G758 is contained G times exactly in this number. Therefore 346 is the quotient of 23382G8 by 6758. 41. The above example worked without omitting the cj'phers would have stood thus : 6758) 2338268 (300 + 40 + 6 2 027400 310868 270320 40548 40548 hence it appears that the divisor is subtracted from the dividend 300 times, and then 40 times from what remains, and then 6 times from what then remains, and there being now no remainder, 6758 is contained exactly 346 times in 2338268. The truth of the above method might have been she^\Ti as follows : 2338268 = 2027400 + 270320 + 40548 67^8) 2027400 + 270320 + 40548 (300 + 40 + G 2027400 + 270320 + 270320 + 40543 + 40548 42. Ex. Divide 56438971 by 4064, 4064) 56438971 (13887 4064 15798 12192 86069 82512 35577 32512 30651 28448 2203 2-5 20 ARITHMETIC. therefore 4064 is contained in 56438971, 13887 times, with the remainder 220a 43. If the divisor terminate with cyphers, the process can he abridged hy the following Rule. Rule. Cut off the cyphers from the divisor, and as many figures from the right-hand of the dividend, as there are cyphers so cut off at the right-hand end of the divisor ; then proceed with the remaining figures according to the Rule, Art. (89) ; and to the last remainder annex the figures cut off from the dividend for the total remainder. Ex. Divide 537528 by 3400. Proceeding by the Rule, 34,00) 5875,28 (158 34 197 170 275 272 3 therefore 3400 is contained in 537523, 158 times with remainder 323. The reason for the Rale will appear from the following considerations. 537523 is 5375 hundreds and 23, of which 537500 contains 3400, 158 times with a remainder 800 over ; and as 23 does not contain 8400 at all, the quotient will evidently be 158, with remainder 800 + 23, or 323. Note. The same rule applies when, the divisor and dividend both terminate with cyphers. 44. Definitions. A number which cannot be separated into factors, which are respectively greater than unity, is called a priaie; number. Thus 3, 5, 7, 11, 18 are prime numbers. A number which can be separated into factors respectively greater than unity, or whicli, in other words, is produced by multiplying together two or more numbers respectively greater than unity, is called a C031P0SITE number. Thus 4 which =2x2, 6 which =2x8, 8 whicli = 2x2x2, are composite numbers; because they are composed or consist of the product of two or mgre numbers, each of which is greater than unity. SIMPLE DIVISION. 21 Numbers which have no common factor greater tlmn unity, are said to be PRIME to one another. Thus the numbers 3, 5, 8, 11, are prime to each other. 45. ^Vhen the divisor is a composite number, and made up of two factors, neither of which exceeds 12, the dividend may be divided by one of the factors in the way of Short Division, and then the result by the other factor : if there be a remainder after each of these divisions, the true remainder will be found by multiplying the second remainder by the first divisor, and adding to the product the first remainder. Ex. Divide 5G732 by 45. ^ rg I 56782 45 < . ^5 6303-5 1260-3 the total remainder is 9 x 3 4- 5, or 27 4- 5 = 32. Therefore the quotient arising from the division of 56732 by 45 i« 1260, with a remainder 32 ovet*. The reason for tJie above Rule is manifest from the following con- siderations. 6303 is 5 times 1260 together with 3, and 56732 is 9 times 6303 together with 5, or is 9 times (5 times 1260 + 3), together wuth 6, or is 45 times 1260 + 27 + 5, or is 45 times 1260 + 32. 46. The accuracy of results in Multij^lication is often tested by the following method, which is termed "castixXg out the nines": add together all the figures in the multiplicand, divide their sum by 9, and set down the remainder ; then divide the sum of the figures in the multiplier by 9, and set down the remainder: multiply these remainders together, and divide their product by 9, and set down the remainder: if this remainder be the same as the remainder which results after dividing the product, or the sum of the digits in the product, of the multiplicand and mul- tiplier by 9^ the sum is very probably right ; but if diiFerent, it is sure to be wrong. This test depends upon the fact that " if any number and the sum of its digits be each divided by 9, the remainders will be the same." The proof of wliich may be shewn thus : 22 ARITHMETIC. 100 = 99 + 1, where the remainder must be one, whether 100, or the sum of the digits in 100, viz. 1, be divided by 9, since 99 is divisible by 9 without a remainder. Similarly, 200 - 2 x 99 + 2, 800 = 3x99 + 3, 400 = 4x99 + 4, 500 = 5x99 + 5, &c. = &c. Hence it appears that if 100, 200, 300, 400, 500, &c. be each divided by 9, and the sum of the digits making up the respective numbers be also divided by 9, the two remainders in each case will be the same. Also the number 532 = 500 + 30 + 2 = 5x100 + 8x10 + 2 =5x99+5+8x9+3+2; whence it appears that if the parts 5 x 100, 3 x 10, and 2, which make up the entire number, be each divided by 9, the remainders will be 5, 3, 2 respectively; and therefore the remainder, when 632 is divided by 9, will clearly be the same, as when 5 + 3 + 2 is divided by 9. To explain why the test holds, let us take as an example 633 multiplied by 67» 583 57 8731 2665 30881 Now 533 = 9x69 + 2 = 531 + 2 67 = 9x 6 + 8= 64 + 8. It is clear, since 531 contains 9 without a remainder, that 531 x 57 contains 9 without a remainder ; therefore the remainder which is left after dividing the product of 683 and 57 by 9, must be the same as the remainder which is left after dividing the product of 2 and 57 by 9. Again, since the product of 57 and 2 = (54 + 8) x 2, and the product of 54 and 2 when divided by 9 leaves no remainder, therefore the remainder which is left after dividing the product of 533 and 57 by 9., SIMPLE DIVISION. 23 must "be tlie same as the femainder left after dividing the product of. 8 and 2 by 9, i. c. after dividing the product of the remaindei-s which are left after the division of the multiplicand and multiplier respect- ively by 9. Now on dividing either 30881, or the sum of its digits, which is 15, by 9, the remainder left is C, and 5x2 divided by nine also leaves 6 as remainder. Therefore we conclude that 80381 is the correct product of 533 and 57. ^ Tsfote, If an error of 9, or any of its multiples, be committed, the results will nevertheless agree, and so the error in that case remains undetected. Ex. VI. Examples in Simple Division. (1) 456-^2. (2) 190680-2, (4) 6378-3. (5) 470850-3. (7) 872096-4. (8) 47392488-4. (10) 9876540-5. (11) 890106-6. (13) 623399-7. (14) (15) 164864-8. (16) (17) 7869231-9. (18) (39) 407792-11. (20) (23) 211632-12. (22) (23) 4045860-13. (24) (2-5) 1234560-20. (26) (27) 14688059-27. (28) (29) 54906734-59. (30) (81) 70865482-87. (82) (33) 28894545-328. (34) (85) 1674918-189. (36) (37) 536819741-907. (88) (39) 8235460800-1440. (40) (41) 353008972662-5 406. (42) (43) 26799534687-7890000. (44) (45) 10000000000000000-1111, and also (46) 634394567-^164600. (47) (48) 1220225292-200563. (49) (50) 60435674586845- f-79094451. (51) (3) 261070308-2. (6) 885784-8. (9) 337625-5. (12) 3782046-6. 78432407-^7. 3812812-8. 89237840 --9. 91875842-11. 43600391 --12. 786543318-4-17. 8224776 -^ 18. 837286228-44. 6848734752 -f- 96. 649305745-^-55. 433438175-^635. 33884740-779. 1111111111311-f- 50160. 57880625 -f- 7575. 599961567212-2468. 57111104051 -f- 3851. by 11111. 67157148372-^90009. 7428927415293 -^ 8496427. 65858547823-^5678. 24 AHITHMETIO. (52) 8968901531C20-f-G87637943. (53) Divide 152181255 by 3854, and explain the proceso. (54) Divide 143255 by 4093. Explain the operation, and shew that it is correct. {55) Divide 203534191 by 72, first by Long Division, and then by its factors 8 and 9 ; and shev/ that the results in both cases coincide. GREATEST COMMON MEASURE. 47. A MEASURE of any given number is a number which will divide the given number exactly, i.e. without a remainder. Thus, 2 is a measure of G, because 2 is contained S times exactly in 6. "When one number is a measure of another, the former is said to measure the latter. 48. A MULTIPLE of any given number is a number which contains it an exact number of times. Thus 6 is a multiple of 2. 49. A coMMox MEASURE of two or more given numbers is a number which will divide each of the given numbers exactly : thus, 3 is a common measure of 18, 27, and 36. The GREATEST COMMON MEASURE of two or more given numbers, is the greatest number which will divide each of tlie given numbers exactly: thus, 9 is the greatest common measure of 18, 27, and 3G. 50. If a number measure each of two others, it will also measure their sum, or difference; and also, any multiple of either of them. Thus, 3 being a common measure of 9 and 15, will measure their sum^ their difference, and also any multiple of either 9 or 15. The sum of 9 and 15 = 9 + 15 = 24 = 3x8; therefore 3 measures their sum 24. The diflference of 15 and 9 = 15 - 9 = 6 = 2 X 3 ; therefore 3 measures their difference 6. Again, 86 is a multiple of 9, and 3G = 3 x 12 ; tlicrefore 3 measures this multiple of 9 ; and similarly any other mul' tiple of 9. Again, 75 is a multiple of 15 ; and 75 = 3 x 25 ; therefore 3 measures tiiis multiple of 15 ; and similarly any other mul- tiple of 15. GREATEST COMMON MEASURE. 25 51. To find the greatest common measure of two numbers. Rule. Divide the greater number by the less; if there be a remainder, divide the first divisor by it ; if there be still a remainder, divide the second divisor by this remainder, and so on ; always dividing the last preceding divisor by the last remainder, till nothing remains. The last divisor will be the greatest common measure re- quired. Ex, Required the greatest common measure of 475 and 589. Proceeding by the Rule given above, 475) 589 (1 475 114) 475 (4 45G 19) 114 (6 114 therefore 19 is the greatest common measure of 475 and 589. Reason for the above process. Any number which measures 589 and 475, also measures their difference, or 589 — 475, or 114, Art. (50), also measures any multiple of 114, and therefore 4 x 114, or 456, Art. (50) ; and any number which measures 456 and 475, also measures their difference, or 475 — 456, or 19 ; and no number greater than 19 can measure the original numbers 589 and 475; for it has just been shewn that any number which measures them must also measure 19. Again, 19 itself will measure 589 and 475. For 19 measures 114 (since 114 = 6 x 19) ; therefore 19 measures 4 x 114, or 456, Art. (50) ; therefore 19 measures 456 + 19, or 475, Art. (50) ; therefore 19 measures 475 + 114, or 589; * . _ therefore since 19 measures them both, and no number greater than 19 can measure them both, C.i? '^ • ' 19 is their greatest common measure. ^ •TT ^ ^ li I - 2G ARITHMETIC. 52. To find the greatest common measure of three or mors numbers. Rule. Find the greatest common measure of the first two numbers ; tlien the greatest common measure of the common measure so found and tlie third number; then that of the common measure last found and the fourth number, and so on. The last common measure so found will he the greatest common measure required. Ex. Find the greatest common measure of 16, 24, and 18, ^Proceeding by the Rule given above, 16) 24 (1 16 8) 16 (2 16 therefore 8 is the greatest common measure of 16 and 24. Now to find the greatest common measure of 8 and 18, 8) 18 (2 16 2) 8 (4 8 therefore 2 is the greatest common measure required. Reason for the above process. It appears from Art. (50) that every number, which measures 16 and 24, measures 8 also ; therefore every number, which measures 16, 24, and 18, measures 8 and 18 ; therefore the greatest common measure of 16, 24, and 18, is the greatest common measure of 8 and 18. But 2 is the greatest common measure of 8 and 18 ; therefore 2 is the greatest common measure of 16, 24, and 18. Ex. VII. 1. Find the greatest common measure of (1)1 6 and 72. (2) SO and 75. (3) 63 and 09. (4) 55 and 121. (5) 128 and 324. (6) 120 and 820. (7) 272 and 425. (8) 894 and 672. (9) 720 and 860. LEAST COMMON MULTIPLE. 27 (10) 825 and 960. (11) (13) 176 and 1000. (14) (16) 089 and 1573. (17) (19) 2023 and 7581. (20) (22) 3444 and 2268. (23) (25) 10395 and 16819. (27) 1242 and 2328. (29) 42237 and 75582. (31) 10353 and 14877. 2. Find the greatest common (12) 856 and 93G. (15) 6406 and 7395. (18) 5210 and 5718. (21) 2484 and 2628. (24) 4067 and 2573. (1) 14, 18, and 24. (2) (3) 13, 52, 416, and 70. (4) (o) 805, 1311, and 1078. (6) (7) 504, 6292, and 1520. (8) 775 and 1800. 1236 and 1632. 1729 and 5850. 468 and 1266. 5544 and 6552. (26) 80934 and 110331. (28) 13536 and 23148. (30) 285714 and 999999. (82) 271469 and 30599. measure of 16, 24, 48, and 74. 837, 1184, and 1847. 28, 84, 154, and 843. 896, 5184, and 6914. LEAST COMMON MULTIPLE. 63. A CoxMMON Mltltiple of two or more given numbers is a number •which, will contain each of the given numbers an exact number of times without a remainder. Thus, 144 is a common multiple of 3, 9, 18, and 24. The Least Common Multiple of two or more given numbers is the least number which will contain each of the given numbers an exact number of times Avithout a remainder. Thus, 72 is the least common multiple of 3, 9, 18, and 24. 54. To find the least common multiple of two numbers. Rule. Divide their product by their greatest common measure : the quotient will b6 the least common multiple of the numbers, Ex. Find the least common multiple of 18 and 80, Proceeding by the Rule given above, 18) 80 (1 18 12) 18 (1 12 6) 12 (2 12 therefore 6 is the greatest comm.on measure of 18 and 80. 28 ABITHMETIO. 18 30 gI540 ~90 therefore 90 is the least common multiple of 18 and 30. Reason for the above process. 18 = 3x6, and 80 = 5x6. Since 3 and 5 are prime factors, it is clear that 6 is the greatest com- mon measure of 18 and 30; therefore their least common multiple must contain 3, 6, and 5, as factors. Now every iniiltiple of 18 must contain 3 and 6 as factors ; and every multiple of 30 must contain 5 and 6 as factors ; therefore every number, which is a multiple of 18 and 30, must contain 3, 5, and 6 as factors ; and the least number which so contains them is 3 x 5 x 6, or 90. Now, 90 = (3 X C) x (5 X 6), divided by 6, = 18x80j divided by 6, = 18 X 30, divided by the greatest common measure of 18 and 30. 55. Hence it appears that the least common multiple of two numbers, wbich are prime to each other, or have no common measure but unity, is their product. 56. To find the least common multiple of three or more numbers. Rule. Find the least common multiple of the first two numbers; then the least common multiple of that multiple and the third number, and so on. The last common multiple so found will be the least common multiple required. ^ Ex. Find the least common multiple of 9, 18, and 24. Proceeding by the Rule given above. Since 9 is the greatest common measure of 18 and 9, their least com* mon multiple is clearly 18. Now, to find the least common multiple of 18 and 24. 18) 24 (1 18 6) 18 (3 18 therefore is the greatest common measure of 18 and 24 ; LEAST COMMON MULTIPLE. 29 therefore the least common multiple of 18 and 24 is equal to (18 x 24) divided by 6, 24 18 192 ' 24 6 I 4,13 72 therefore 72 is the least common multiple required. Reason for the above process. Every multiple of 9 and 18 is a multiple of their least common multi- ple 18; therefore every multiple of 9, 18, and 24 iS a multiple of 18 and 24 ; and therefore the least common multiple of 9, 18, and 24 is the least com- mon multiple of 18 and 24: but 72 is the least common multiple of 18 and 24 ; therefore 72 is the least common multiple of 9, 18> and 24. 67. When the least common multiple of several numbers is required, the most convenient practical method is that given by the following Rule. Rule. Arrange the numbers in a line from left to right, with a comma placed between every two. Divide those numbers which have a common measure by that common measure, and place the quotients so obtained and the undivided numbers in a line beneath, separated as before. Proceed in the same way with the second line, and so on with those which follow, until a row of numbers is obtained in Avhich there are no two numbers which have any common measure greater than unity. Then the continued product of all the divisors and the numbers in the last line will be the least common multiple required, JVote. It will in general be found adv^antageous to begin with the lowest prime number 2 as a divisor, and to repeat this as often as can bo done ; and then to proceed with the prime numbers 3, 5, &c. in the same way. Ex. Find the least common multiple of 18, 28, 30, and 43. Proceeding by the Rule given above, 2 18, 28, 30, 42 2 9, 14, 15, 21 8 9, 7,15,21 7 3, 7, 5, 7 3, 1, 5. 1 so ARITHMETTC. therefore the least common multiple required = 2x2x3x7x3x5 = 1269. Reason for the above process. Since 18 = 2x3x3; 28 = 2x2x7; 80 = 2x3x5; 42 = 2x3x7; it is clear that the least common multiple of 18 and 28 must contain as a factor 2x2x8x3x7; and this factor itself is evidently a common multiple of 2 X 3 X 3, or 18, and of 2 x 2 x 7, or 28 ; now the least number which contains 2x2x3x3x7 as a factor, is the product of these numbers; therefore 2x2x3x3x7 is the least common multiple of 18 and 28 : also it is clear that the least common multiple of 18, 28 and 30, ©r of 2x2x3x3x7 and 30, or of 2x2x3x3x7 and 2x3x5 must contain as a factor 2x2x3x3x7x5, and this factor itself is evidently a common multiple of 2 x 3 x 3 or 18, 2 x 2 x 7 or 28, and 2 x 3 x 5 or 80 ; hence it follows as before that 2x2x3x3x7x5 is the least common multiple of 18, 28, and 30 ; again tlie least common multiple of 2x2x3x3x7x5 and 42, or of 2 x 2 x 3 x 3 x 7 x 5 and 2x3x7 must contain 2x2x3x3x7x5 as a factor, and this factor, as before, is evi- dently itself a common multiple of 18, 28, 30, and 42 ; now the least number which contains 2x2x3x3x7x5 as a factor, is the product of these numbers. Therefore this product, or 1260, is the least common multiple re- quired. Note 2. The above method is sometimes shortened by rejecting in any line, any number, which is exactly contained in any other number in the same line ; for instance, if it be required to find the least common multiple of 2, 4, 8, 16, 10 and 48 ; the numbers 2, 4, 8, 16, since each of them is exactly contained in 48, may be left out of consideration, and 240, the least common multiple of 10 and 48, will evidently be the least common multiple required. Ex. VIII. 1. Find the least common multiple of (1) 16 and 24. (2) 36 and 75. (3) 7 and 15. (4) 28 and 35. (5) 319 and 407. (6) 833 and 504. (7) 2961 and 799. (8) 7568 and 9504. (9) 4662 and 5470. (10) 6327 and 28997. (11) 5415 and 80105. (12) 15863 and 21489. MISCELLANEOUS QUESTIONS. 31 2. Find the least common multiple of (1) 12, 8, and 9. (2) 8, 12, and IG. (3) G, 10, and 15. (4) 8, 12, and 20. (5) 27, 24, and 15. (6) 12, 51, and 68. (7) 19, 29, and 38. (8) 24, 48, 64, and 192. (i» 63, 12, 84, and 14. (10) 5, 7, 9, 11, and 15. (11) 6, 15, 24, and 25. (12) 12, 18, 30, 48, and 60. (13) 15, 35, G3, and 72. (14) 9, 12, 14, and 210. (15) 64, 81, 63, and 14.' (16) 24, 10, 82, 45, and 25, (17) 1,2,3, 4, 5, 6, 7, 8, and 9. (18) 7,8,9, 18 !, 24, 72, and 144. (19) 12, 20, 24 , 54, 81, 63, and 14. (20) 225,255,; 289, 1023, and 4095. Ex. IX. Miscellaneous Questions and Examples on the foregoing Articles. L (1) Explain the principle of the common system of numerical notation. "Multiply 603 by 48, and give the reasons for the several steps. (2) Write at length the meaning of 9090909, and of 90909. Find their sum and difference, and explain fully the processes employed. (3) Find the difference between the sum of 4715 added to itself 898 times, and the sum of 2017 added to itself 408 times. (4) A person, whose age is 78, was 37 years old at the birth of his eldest son ; what is the son's age ? (5) Explain the meaning of the terms 'vinculum*, 'bracket'; and of the signs +, — , =,.*., x. Find the value of the following expression : 15x37153-78474-671524-4 + 40734x2. II. (1) Define 'a Unit', 'Number', 'Arithmetic'. What is the difference between Abstract and Concrete numbers ? (2) The annual deaths in a town being 1 in 45, and in the country 1 in 50; in how many years will the number of deaths out of 18675 persous living in the town, and 79260 persons living in the country, amount together to 10000 ? (8) Define 'Notation', 'Numeration'; express ia numbei-s seven hundred thousand four hundred and nine billions. '32 ARITHMETIC. (4) Find the value of 494871 -04853+(45079-3l77)-(64312-S987)-(1763 + 231)f 379x379. (5) What number divided by 528 will give 36 for the quotient, and leave 44 as a remainder 1 III. (1) Define Multiplication, and Division. Shew that the product of two numbers is the same in whatever order tlie operation is performed. (2) The Iliad contains 15683 lines, and the ^neid contains 9892 lines ; how many days will it take a boy to read tlirough both of them, at the rate of eighty-five lines a day ? (3) Explain what is meant by the greatest common measure, and by the least common multiple of two or more numbers ; and shew that the product of two numbers is the product of their least common multiple into their greatest common measure. Find the least common multiple of 22, 16, 21, 52, and 70. (4) Explain the meaning of the sign -f-, and find the value of (7854-4913) x3-(20374-12530)-53-6 + (395456-2S64)^556. (5) At a game of cricket J, B, and C together score 108 runs; B and C together score 90 runs, and A and C together score 51 runs ; find the number of runs scored by each of them. IV. (1) Define Addition, and Subtraction. What is meant by a prime number? When are numbers said to be prime to each other? Give examples. Explain the rule of carrying in the addition of numbers ; exemplify it in the addition of 8864, 4768, and 15988. (2) There are two numbers of which the product is 373625 ; the greater number is 875 ; find the sum and difference of the numbers. (3) A father was 21 years old when his eldest son was born ; how old will his son be when he is 50 years old, and what will be the father's age when the son is 50 years old ? (4) Write in figures one hundred millions, one hundred thousand, one hundred and one; and in words 1010101010. Express in figures M.DCCC.XL. (5) When are numbers said to be ' composite' ? Find the greatest number which can divide each of the two numbers 849 and 1132; also the least number which can be divided by each of them ; explaining the process in each case. MISCELLANEOUS QUESTIONS. 33 V. (1) Multiply 478 by 146, and test the result by casting out the nines. In what cases does this method of proof fail ? Divide 4843 by 99, and prove the correctness of the operation by any test you please. (2) ^Vhat number multiplied by 8G will give the same product as 163 by 430? (8) In the city of Prague, for every two persons who speak German only, three speak Tschech onh'-, and seven both German and Tschech ; and the whole population is 120000. How many speak German only, Tschech only, and both German and Tschech ? (4) A gentleman dies, and leaves his property thus : 10000 pounds to his widow ; 15000 pounds to his eldest son, on the condition of his building a national school at a cost of 350 pounds; 5500 pounds to each of his four younger sons ; 3750 pounds to each of his three daughters ; 4563 pounds to oifFerent societies; and 599 pounds in legacies to his servants. What amount of property did he die possessed of? (5) The quotient arising from the division of 9281 by a certain number is 17, and the remainder is 873. Find the divisor. VI. (1 ) Explain briefly the lloman method of Notation. Express 15C3 and 9000 in Roman characters. (2) Explain the terms 'factor', "^product', 'quotient'; shew by an example how the process of Division can be abridged, if the divisor terminate with cyphers. (3) The remainder of a division is 97, the quotient 665, and the divisor 91 more than the sum of both. What is the dividend ? (4) Express in words the numbers 270130 and 26784; also write do:vn in figures the number ten thousand, two hundred and thirty four ; and find the least number which added to the last number will make it divisible by 8. (5) A gentleman, whose age is 60, has two sons and a daughter; his age equals the sum of the ages of his children ; two years smce his age was double that of his eldest son ; the sum of the ages of the father and the eldest son is seven times as great as that of the yomigest son ; find the ages of the children. 34 ARITHMETIC. FRACTIONS. 58. If 1 represent any concrete quantity, as for instance 1 yard, it is divisible into parts : suppose the parts to be equal to each other, and the number of them 3; one of the parts would be denoted by |- (read one-third), two of them by f (read two-thirds), three of them or the whole yard by f or 1 j if another equal portion of a second yard divided in the same manner as the first be added, the sum would be denoted by f ; if two such portions were added, by # ; and so on. Such expressions, representing any number of pai'ts of a unit, that is, of the quantity which is denoted by 1, are termed Broken Numbers or Fractions ; we may therefore define a fraction thus : 59. Def. a Fraction denotes a part or parts of a unit ; it is ex- pressed by two numbers placed one above the other with a line drawn between them ; the lower number is called the Denominator, and shews into how many equal parts the unit is divided ; the upper is called the Numerator, and shews how many of such parts are taken to form the fraction. Thus f denotes that the unit is divided into 6 equal parts, and that 5 of these parts are taken to form the fraction : so, if a yard were divided into 6 equal parts, and 5 of them were taken, then denoting one yard by 1, we should denote the parts taken by the fraction |. Again, ^ denotes that the unit is divided into 6 equal parts, and that 7 such parts are taken to form the fraction ; for instance, in the example before us, one whole yard would be taken, and also one of the equal parts of another yard divided in the same manner as the first. 60. A Fraction also represents the quotient of the numerator by the denominator. Thus, f represents 5 -f- 6; for we should obtain the same result, whether we divide one unit into 6 equal parts, and take 5 of such parts (which would be represented by |) ; or divide^re units into 6 equal parts, and take 1 of such parts, which would be equivalent to ^*^ part of 5 units, t. e. 5 -r 6 : hence f and 5 -f- 6 will have the same meaning. 61. When fractions are denoted in the manner above explained, they are called Vulgar Fractions. Fractions, whose denominators are composed of 10, or 10 multiplied VULGAR FRACTIO:>'S. 35 by itself, any number of times, are often denoted in a different manner ; and when so denoted, they are called Decibial Fractions. VULGAR FEACTIONS. 62. In treating of the subject of Vulgar Fractious, it is usual to make the following distinctions : (1) A Proper Fraction is one whose numerator is less than the denominator; thus, f, f-, f- are proper fractions. (2) An I3IPB0PER Fraction is one whose numerator is equal to or greater than the denominator ; thus, f, f , f are improper fractions. (3) A Simple Fraction is one whose numerator and denominator are simple integer numbers ; thus, ^, ^ are simple fractions. (4) A Mixed Number is composed of a whole number and a fraction ; thus, 5}, 7f are mixed numbers, representing respectively 5 units, together with Jth of a unit ; and 7 units, together with £ths of a unit. (5) A Compound Fraction is a fraction of a fraction; thus, ^ of £, 1^ of I" of ^Q are compound fractions. (G) A Complex Fraction is one which has either a fraction or a mixed f 2\ 8 21 number in one or both terms of the fraction; thus, -j, -^ ^ ^, -|, - - are complex fractions. 63. It is clear from what has been said, that every integer may be considered as a fraction whose denominator is 1 ; thus, 5 = f, for the unit is divided into 1 part, comprising the whole unit, and 5 of such parts, that is 5 units, are taken. 64. To multiply a fraction by a whole number , multiply the numerator of the fraction by it. Thus, f x3 = f. Jleasonfor the above process. In f- the unit is divided into 7 equal parts, and 2 of those parts are taken : whereas in f- the unit is divided into 7 equal parts, and 6 of those parts are taken; i.e. 3 times as many parts are taken in f as are taken in f, the value of each part being tlio same in each case. 3—2 86 ARITHMETIC. Ex. X. (1) Multiply j% separately by S, 9, 12, 3G. (2) Multiply Ai- separately by 7, 15, 21, 45. 65. To divide a fraction by a whole number, multiply the denominator by it. r^^ 2 . ^ 2 2 Reason for the above process. In the fraction f , the unit is divided into 7 equal parts, and 2 of those parts are taken ; in the fraction ^\, the unit is divided into 21 equal parts, and 2 of such parts are taken : but since each part in the- latter case is equal to one-third of each part in the former case, and the same number of parts are taken in each case, it is clear that ^ represents one-third part off, or f-^3. Ex. XI. (1) Divide f separately by 2, 3, 4, 5, 10. (2) Divide -gi^ separately by 11, 20, 25, 45. 66. If the numerator and denominator of a fraction be both multiplied or both divided by the same number , the value of the fraction will not be altered. Thus, if the numerator and denominator of the fraction f be multiplied by 8, the fraction resulting will be ^, which is of the same value as f. Heason for the above pi'ocess. In the fraction ^ the unit is divided into 7 equal parts, and 2 of those |)arts are taken ; in the fraction ^ the unit is divided into 21 equal parts, and 6 of such parts are taken. Now there are 3 times as many parts taken in the second fraction as there are in the first fraction; but 3 parts in the second fraction are only equal to 1 part in the first fraction ; therefore the 6 parts taken in the second fraction equal the 2 parts taken in the first fraction ; therefore f — -^. 67. Hence it follows that a whole number may be converted into a vulgar fraction with any denominator, by multiplying the number by the required denominator for the numerator of the fraction, and placing the required denominator underneath ; for 6=5-5 VULGAR FRACTIONS. 37 and to convert it into a fraction with a denominator 5 or 14, we have 6 1 6 "1 x5 x5 80 5^ 6 l' 6 ~1 xl4 xl4 84 "14* Ex. Xil Reduce (1) 7, 0, and 11, to fractions with denominators 3, 7, and 22 respectively ; and (2) 26, 109, 117, and 125, to fractions with denominators 2, 5, 13, 23, and 85 respectively. 68. Multiplying the numerator of a fraction by any number, is the same in effect as dividing the denominator by if, and conversely. For if the numerator of the fraction f be multiplied by 4, the resulting fraction is ^ j and if the denominator be divided by 4, the resulting fraction is f . Now the fraction ^/ signifies tliat unity is divided into 8 equal parts, and that 24 such parts are taken ; these are equivalent to 3 units : also f signifies that unity is divided into 2 equal parts, and that 6 such parts are taken ; these are equivalent to 3 units : hence ^ and f are equal. The proof of their equaUty may also be put in this form : that since the unit, in the case of the second fraction, is only divided into 2 equal paiis, each part in that case is 4 times as great as each part in the case of the first fraction, where the unit is divided into 8 equal parts ; and therefore 4 parts in the case of the first fraction are equal to 1 part in the case of the second ; or the 24 parts denoted by the first are equal to the 6 denoted by the second ; or, in other words, the fractions ^^ and f are equal. Again, if we divide the numerator of the fraction f by 2, the resulting fraction is | ; and if we multiply the denominator by 2, the resulting fraction is ^-^. Now, f signifies that the unit is divided into 8 equal parts, and that 3 of such parts are taken ; and j^ signifies that the unit is divided into 16 equal parts, and that 6 of such parts are taken : but each part in | is equal to 2 parts in ^^ ; and therefore | is of the same value as - , or — . 69. To represent an improper fraction as a whole or mixed number. Rule. Divide the numerator by the denominator : if there be no remainder, the quotient will be a whole number ; if there be a remainder. 38 AHITHMETIC. put down the quotient as the integral part, and the remainder as the numerator of tlie fractional part, and the given denominator as the deno- minator of the fractional part. Ex. Reduce "-^ and ^^ to whole or mixed nurahers. By the Rule given above, %^=5, a whole number ; 6 ^6' Reason for the above process. Smce —=:——-=- X 5, (Art. 64), o o «5 and since |- signifies that the unit is divided into 5 equal parts, and that 5 of those parts are taken, which 5 parts are equal to the whole unit or 1 ; therefore "/ =f x 5 = 1 x 5, or 5. J, . So 80 + 5 6x5 + 5 Agam, ^ = __ = — ^ , 6x5 which equals —^ together with f , that is, = 5 together with f, by what has been said above ; or, as it is written, 5|. Ex. XIII. Express the following improper fractions as mixed or whole numbers : (1) ¥• (2) y. (3) ¥• (4) ^^^- 183 55 • (5) «^. (6) W. (7) W- (8) (9) W' (10) ^F- (11) f§f- (12) Wt'- (13) LO^?A (14) s^Y^yj). (15) ii|-|4. (16) 9^5^. (17) iwi. (18) ^mi^- (19) wm- (20) m¥. 70. To reduce a mixed number to an improper fraction. Rule. Multiply the integer by the denominator of the fraction, and to the product add the numerator of the fractional part ; the result will be the required numerator, and the denominator of the fractional part the required denominator. Ex. Convert 2f into an improper fraction. Proceeding by the Rule given above, 2x7 + 4 10 ^f n -7 • VULGAR FRACTIONS. 39 Reason for the above process. 2f is meant to represent the integer 2 with the fraction f- added to it. 2x7 14 But 2 is the same as — =— or -=-; and therefore 2f must be the same as Y increased by f , or as V^ : for '^^ denotes that unity is divided into 7 equal parts, and represents 14 such parts together with 4 such parts. Ex. XIV. Reduce the following mixed numbers to improper fractions : (1) 21 (2) 5f. (3) 4^. (4) 7t. (5) 2511. (6) 43iV (7) 25j\. (8) 14if. (9) 20031 (10) 857^. (11) 57e. (12) ISff. (13) 3^. (14) 26|M. (15) 164iif. (16) 106|il. (17) 157i§t. (18) irUU' (19) 427r3^. (20) lOOfi^. 71. To reduce a compound fraction to its equivalent simple fraction. Rule. ]\Iuitiply the several numerators together for the numerator of the simple fraction, and the several denominators together for its denominator. Ex. Convert f of |- into a simple fraction. Proceeding by the Rule given above, 3 ^7^3x7^21 5 ^^ 8 ~ 6 X 8 ~ 40 * Reason for the above process. By f of I", we mean §ths of that part of unity which is denoted by I : thus if unity be divided into 8 equal parts, and 7 of these be taken, and if each of these be again divided into 5 equal parts, and 3 of each set of parts be taken, then each of the parts will be one-fortieth part of the original unit, and the number of parts taken will be 3 x 7, or 21 ; 21 3x7 the result therefore is — , or ~-^ ; that is, 3 . 7^3x7 5 8 5x8' JVote. In reducing compound fractions to simple ones, we may strike out factors common to one of the numerators and one of the 40 ARITHMETIC. denominators : for this is in fact simply dividing the numerator and denominator of the fraction by the same number. Art. (66)^ Thus f of 2 j\ of 1-^ = f of f I of -]^ _ 8 X 25 X 10 _ 3 X 5 X 5 X 4 X 4 ~5x 12 X 15 "5x3x4x8x5 Cstriking out the factors S, 5, 5, 4 from the numerator and do- nominator) Ex. XV. Reduce the following- compound fractions to simple ones : (1) -I off. (2) -?of^. (3) foff. (4) fofll. (5) I off of 7. (6) foffoftofJ^of28. (7) y\ of 21 off of 101 (8) ^ of 121 of A of foff of 9. (0) ^Vof^o>f§off of,2_of2'^of^. (10) f off off of70tof4^ofl/3-ofl47. 72. Def. a Fraction is in its lowest terms, wdien its numerator and denominator are prime to each other. Note. When the numerator and denominator of a fraction are not prime to each other, they have (Art. 44) a common factor greater than unity. If we divide each of them by this, there results a fraction equal to the former, but of wliich the terms, that is, the numerator and denommator are less, or lower than those of the original fraction ; and it may be considered to be the same fraction in lower terms. When the numerator and denominator of a fraction are prime to each other, that is, have no common factor greater than unity, it is clear that its terms cannot be made lower by division of this kind, and on this account the fraction is said to be in its lowest terms. 73. To reduce a fraction to its lowest terms. Rule. Divide the numerator and denominator by their greatest common measure. Ex. 1. Reduce ff|| to its lowest terms. First, find the greatest common measure of 6465 and 7335. VULGAR FRACTIONS, 41 G465) 7835 (1 G465 070) 6465 (7 6090 '375) 870 (2 750 120) 875 (3 860 15) 120 (8 120 therefore 15 is the greatest common measure. 15) 6465 (431 15) 7835 (489 60 60 46 133 45 120 15 135 15 135 therefore fraction m its lowest tenns = |-|i. Reason for the above process. If the numerator and denominator of a fraction be divided by the same number, the value of the fraction is not altered (Art. (jG) ; and the great- est number which will divide the numerator and denominator is their greatest common measure. Note. Sometimes it is unnecessary to find the greatest common measure, as it is easier to bring the fraction to its lowest terms by successive divisions of the numerator and denominator by common factors, which are easily determined by inspection. Ex. 2. Reduce f f§ to its lowest terms, 5-|o _ 6|.^ dividing numerator and denominator by 10, = If, dividing numerator and denominator by 3. Ex. XVI. Reduce each of the following fractions to its lowest terms : (1) |. (2) ft. (3) M. (4) (6) U' (6) It. (7) i%v (8) m- 13. 42 ARITHMETIC. (9) III. (10) m- (11) m- (12) ffg. (18) ^m- (14) M. (15) m- (16) ^^. (17) fHf. (18) mk' (19) oW^. (20) Me. (21) ^^^. (22) ^|i^. (23) Mf|. (24) Ifm. (25) /A^V. (26) MMi. (27) |-?f|. (28) i^^. (29) Mm. (80) %m^t' (81) M^Hf. (82) MIMf. 74. To reduce fractions to equivalent ones with a common deno- minator. Rule. Find the least common multiple of the denominators : this will be the common denominator. Then divide the common multiple so found by the denominator of each fraction, and multiply each quotient so found into the numerator of the fraction which belongs to it for the new numerator of that fraction. Note 1. If the given fractions be in their lowest terms, the above rule will reduce them to others having the least common denominator; if the least common denominator be required, the given fractions should be reduced to their lowest terms before the rule be applied^ Ex. Reduce j|-, ^^, i|-, |^, into equivalent fractions with a common denominator. Proceeding by the Rule given above. 2 12, 16, 24, S3 2 6, 8, 12, 83 2 8, 4, 6, 83 8 8, 2, 3, 83 1, 2, 1, 11 . therefore least common multiple =2x2x2x3x2x11 = 528; therefore the fractions become respectively, 5 X 44 220 12 X 44 528 9 X 88 297 / . 528 f' 528 .A (since — =44 j, 297 / . 528 ^_\ 16x83 = 528V^^^^^l6- = ^^;' 11x22 242/. 528 \ 24722=528^^^^-24=^7' 17 X 16 272 / . 528 33x16 272/. 528 ,A = 5-28V^^^^83=^^j' VULGAR, FRACTIONS. 43 or the fractions with a common denominator are Z50. SOJL SJL3. 2JL2 r>-ZS> 6iib» 628' 62¥" licas'on for the above process. The least common multiple of the denominators of the given fractions will evidently contain the denominator of any one of the fractions an exact numhcr of times. If both the numerator and denominator of that fraction be multiplied by that number, the value of the fraction will not be altered (Art. (jQ) ; and the denominator will then be equal to the least common multiple of all the denominators. If this be done with all the fractions, they will evidently be, in like manner^ reduced to others of the same value, and having the least common multiple of all the denominators for the denominator of each fraction. Note 2. If the denominators have no common measure, we must then multiply each numerator into all the denominators, except its own, for a new numerator for each fraction, and all the denominators together for the common denominator. Ex. Reduce \, f , |-, to equivalent fractions with a common deno- minator. The least common multiple of the denominators = 5 X 7 x9 ; therefore the fractions become 1 X 7 X 9 _ 63 5x7x9~3]5' 2x5x9 _ 90 7x5x9~8J5' Ix5x 7_ 35 9 X 5 x 7 ~ 815 ' or the fractions with a common denominator are ^\5) ^TS) '''"^ 315* Ex. XVII. Reduce the fractions in each of the following sets to equivalent frac- tions, having the least common denominator : (1) \, i and f . (2) f , and |. (3) %, f , and |. (4) f , and ^. 44 ARITHMETIC. (5) (7) (9) (11) h Ti> and iJ-. I, A, and H. h h h and |. I, I, and |. j^, ^, and i|. 7 (IS) i M,and^-,%. (15) ^, i^. ]h ^, and .V. (17) h h h> -k^ ^%> and ,^^. (19) t^-, ^S-, Ih T^, and f. (6) (8) (10) (12) (14) tV, f. ii,and-H_. (16) i I, f , i^T. I^. and ||. (18) (20) 7» Tx> iii> and f . 9_ 9 2___ nnrl 0, 10) 3 00^ 1000? '^^^^ 10000' 3JL U. 5j5 r>n(J ^3_ 5 4-' 2S' (To? "■^^^ 12* N'ote 3. Whenever a comparison has to be made between fractions, in respect of their magnitudes^ they must be reduced to equivalent ones with a common denominator ; because then we shall have the unit divided, in the case of each fraction so obtained, into the same number of equal parts ; and the respective numerators will shew us how many of such parts are taken in each case ; or which is the greatest fraction, which the next, and so on. Ex. Compare the values of i,\, ^l, §, ^, and f . First, to find the least common multiple of the denominators ; 2 27, 24, 6, 15, 5 8 27, 12, 8, 15, 5 5 0, 4, 1, 5, 5 9, 4, 1, 1, 1 therefore the least common denominator =2x8x5x9x4 = 1080; therefore the fractions become 5x40 200 27x40 '1080' 11x45 495 24 X 45 ~ ' 1080^ 5x180 900 6x180 1080' 4x72 288 16x72 '1080' ADDITION OF VULGAR FRACTIONS. 45 Sx216_ G48_ 5 X 216 ~ 1080' therefore f is the greatest;, § the next, l^ the next, -^^ the next, and -^ the least. Ex. XVIII. 1. Compare the values of (1) (2) (3) (4) if. and, V h, f, f, and |. i off,/., and 4 off. j%, fo. if' and fl. (5) i 1^, WV, t\. and e. (6) f of I of 4, ^j of f of 5, i- of -I of 4J, and ^ . (7) i M' ^' tV. and f^. (8) V.S^andfofGt. (S) f,M,liiandf|. no^ ^ -5 7_ JL and ^ ni^ -il U-3. 1-1- 40X and -^-20 \^^>' 76' 152' -^as' 44S. '^"'"^ 748" (12) V, 3^, t of 9f, and toff off. Find the greatest and least of the fractions (1) i T2' h h and i (2)-H4I.H.lV.and|-|. ADDITION OF VULGAR FRACTIONS. 7o. Rule. Reduce the fractions to equivalent ones with their least common denominator ; add all the new numerators together, and under their sum write the common denominator. Ex. Find the sum of -- , — , and ^-/ . lo 21 ' 35 Proceeding by the Rule given above, First, find tlie least common multiple of the denominators ; 8 15, 21, 35 5 5, 7, 35 7 1, 7, 7 1, 1, 1 46 ARITHMETIC. therefore the least common multiple =3x5x7 = lOd 3 therefore the fractions become 7x7 _ 49 15x7~105' 10 X 5 _ ^ 21x5" 105' 16x3 48 therefore their sum = 35 x 8 105 ' 49 + 50 + 48 147 105 105' _49 ~35' Reason for the Rule. — ^s' In each of the equivalent fractions, we have unity divided into 105 equal parts, and those fractions represent respectively 49, 50, and 48 of such parts ; therefore the sum of the fractions must represent 49 + 50 + 48 147 or 147 such parts, that is, must be r^ . IVote 1. If the sum of the fractions be a fraction which is not in its lowest terms, reduce it to its lowest terms ; and if the result be an improper fraction, then reduce it to a whole or mixed number: thus ]-^ = si — ^M • the same remark applies to all results in Vulgar Fractions. JVote 2. Before applying the rule, reduce all fractions to their lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones. Note 3. If any of the given numbers be whole or mixed numbers ; the whole numbers may be added together as in simple addition, and the fractional parts by the Rule given above. 8 9 Ex. Find the sum of - , 3}4, lOf , and ^ . 8 14 2 9 Now to find the sum of- + — + ^ + ^. 8 15 o Z^ ADDITION OF VULGAR FRACTIONS. 47 First, find the least common multiple of the denominators ; 2 I 8, 15, 5, 22 5 4, 15, 5, 11 4, 8, 1, 11 therefore the least common multiple = 2x5x4x3x11 = 1320; therefore the fractions become 8 X 165 495 8 X 165 1320 * 14 X 88 1232 15x88 1320' 2 X 264 528 6x264 "1320' 9 X 60 540 22x60" 1320' therefore the sum of the fractions 495 + 1232 + 628 + 540 1820 2795 ~ 1320 - ?^^ , dividing numerator and denominator by 5, = 2,%; therefore the whole eum -13 + 2|^, = 15^. Ex. XIX. 1. Add together, (1) fandf (2) fandf (3) (4) ^and,^. (5) t\and^V (6) (7) j^and/3. (8) ^andfi. (9) (10) i^andj^. (11) 8fand7t. (12) fandf. A and If. f and 21. 4|- and 9-}|. 2. Find the sum of (1) |,f,and^. (2) if,and^ • 48 AEITHMETIC. (8) lf,and3i,. (4) A^ i and ^. (5) if,and/^. ' (6) f , f , and l. (7) |,f,andtV (8) i^,landl (9) f,3^,and^V (10) f, 21 and 183^. i h and /o' h h and TTx* h r. 45 -, and>^. h f' of^, andO^^y. h h h and ^J-. h To> h and ^. (11) ifofi,and9^^. (12) ^ of f of i 51 and t^. (13) |,A^,and^l (14) la ^, and la. (15) f,/o,t,and^. (16) 1 ^, f, and ^. (17) i6f,andiof^. (18) lOOf, 64|, | of 701. (19) 261^, I74f , and f of 10|. (20) 887^,2851 8941, and f of 8704. 3. Find the value of (1) H + ^V + li^ + To^- (2) H + 11 + M + tl + M. (3) ^ + i + ^ + f|+|f. (4) 2ii + 6|l + if + ^ofif + M+|of2f. (6) 2f+8f + 4l + 5H6f (6) li + 8J^ + i|4-7A + ^+rofi (7) 51 + 4 of f- of 81 + 93^+^ off of 4. (8) A of 12 + t of f isf of If of J^V + if of 8J of g\ of 1-^-, (9) 270f + GoO-1j + 50001 + 58^ + 1J^). (10) iofi + //of(l4-if) + IJ + |-fofil + i|. SUBTRACTIOK 76. Rule. Reduce the fractions to their least common denominator, take the difference of the new numerators, and place the common denomi- nator underneath. 1 7 Ex. Subtract -^ from -g . Proceeding by the Rule given above, since 8 is clearly the least common multiple of the denominators, the equivalent fractions will be I and I, 7 — 4 3 and their difference = — — = g . Reason for the Ride. The unit in each of the equivalent fractions is divided into 8 equal parts^ and there are 7 and 4 parts respectively taken, and therefore tho SUBTRACTION OF VULGAR FRACTIONS. 49 diflFerence must be 3 of such parts, or, in other words, the diflferencc of the two fractions is f . Note 1. Remember always, before applying the above Rule, to reduce fractions to their lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones. Note 2. If either of the given fractions be a whole or mixed number, it is most convenient to take separately the difference of the integral parts and that of the fractional parts, and then add the two results together, as in the following examples. Ex. 1. From 4| subtract 1\. Here 4-2 = 2, and |-i = |-| = |; therefore the difference of 4| and 1\ — 1\. For the process expressed at length is 4-l-|-(2 + i). which =4 + 1-2-1 Art. (1 2), or =4-2 + (|-i) Ex. 2. Take 2| from 4J. Now f cannot be taken from \, since it is the greater of the two ; we therefore add 1 to \, and take % from 1 + 1 or | ; and then, in order that the difference may not be altered, we add 1 to the 2. ■l-^OW 4_ 8 8 8 ~S' 4-3=1; therefore the difference of 4} and 2^ = 1|-. For the process expressed at length is 4 + i-(2 + |) which =4 + 1+}- (2 + 1 + 1) (adding and subtracting 1), = 4 + |-(3 + |) =4-3+1-1 ^i + V'-l = 1 + 1 = 1|. Ex. XX. 1 . Find the difference between (1) f and 1. (2) 1 and \. (8) A and ^. (4) ^ and -f-^. (5) 11 and H- (6) Aand^. (7) 2f and \\. (8) 87A and 38./^. (0) 6| and ^, (10) 13^^ and 9/^. (11) fiOiV and 47^. (12) 42 and 30i%, 4 50 ARITHMETIC. (13) 15^% and UYij- 04) DO^^'^^ nnd 25-/^^ (15) 21 and li||. (16) 125 and f of 14, (17) 46|.andl5i (18) ifand^^^ofli (19) |of|-off andf off. (20) f of f of f of 8| and ^ of f of ^ of l^y. 2. By how much does | of ^^— | of ^ exceed f of ^— | of j^ ? ?. Add j^ of f to 2| and subtract f; from the result. 4. From the sum of 11| and 8-^- subtract 9?r|. 5. By how much does the difference of 5p- and 2f exceed the sum 6. By how much does the sum of the fractions Ig^g- and ^% exceed their difference ? MULTIPLICATIOK 77. Rule. Multiply all the numerators together for a new nume- rator, and all the denominators together for a new denominator. Ex. Multiply I by I . Proceeding by the Rule given above. 3^5 15 7x8 ~ 56* Reason for the Rule, If f be multiplied by 6, the result is V, Art. (64). But this result mast be 8 times too large, since, instead of multiplying by 5, we have only to multiply by |, which is 8 times smaller than 5, or, in other words, is one-eighth part of 5. Consequently the product above, viz. Y ii^ust be divided by 8, and Vh-8=^, Art. (65). Note 1. The same rcasonmg will apply, whatever be the number of fractions which have to be multiplied together. Note 2. Before applying the above Rule mixed numbers must be reduced to improper fractions. Note 3. It has been shewn that a fraction is reduced to its lowest terms by dividing its numerator and denominator by their greatest common measure, or, in other words, by the product of those factors which are common to both ; hence, in all cases of multiplication of fractions, it will be well to split up the numerators and denominators as much as possible into the factors which compose them; and then, after MULTIPLICATION OF FRACTIONS, 51 putting the several fractions under the form of one fraction, the sign of X being placed between each of the factors in the numerator and denominator, to cancel those factors which are common to both, before carrying into effect the final multiplication. Thus, in the following Examples : 3 4 Ex. 1. Multiply - and -z together. Product ='. — -, 4x5 Now cancelling, i.e. dividing the numerator and denominator by the common factor 4, we see that product = '- . o 12 3 Ex. 2. Multiply ^ » o" » ^^^ I together. „ , ^ 1x2x3 rroduct =- — - — ; , 2x3x4* (or cancelling, i.e. dividing numerator and denominator by the product of the common factors 2, 3,) ^1 ~4* 8 16 27 45 60 8 X 16 X 27 X 45 Ex. 3. Multiply 9 » ^ ^ gQ . ^ together. Product = 9x24x80x60' 8x4x4x3x9x5x9 9x8x8x5x6x6x12* (or cancelling, i. e. dividing the numerator and denominator by tha product of the common factors 8, 3, 9, 5,) , , 4x4x9 P""^^^^'* =^6x5x12' 4x2x2x3x3 "3x2x6x3x4 (or cancelhng, i.e. dividing numerator and denominator by the product of the common factors 4, 2 3, 3,) 2 product in its lowest terms =- . Ex. 4. Multiply 2 1 , 3 1 , 10 ^ , 20 1 , and 6 ^ together. 4—2 52 ARITHMETIC. 13 14 9 2 x3-x 10- X 20^^x523, 5 27 81 184 124 ""2''8'^8'' 9 "^23' _5x9x3x9x9x8x23x4xSl ~ 2x2X4k8K95<23 ' (or cancelling, i. e. dividing numerator and denominator by the prodr ct of the commoti factors 9, 8, 23, 4,) 5x3x9x9x31 product 2x2 _ 37865 ~ 4 ' = 9416^. Ex. XXI. 1. Multiply (1) -fby|. (2) ebyH. (3) f-byf. (4) i^by^. (5) f^byif. (6) 7^ by J. (7) 8fby2i-. (8) 71- by 1 off. (9) 12 by | of 5. (10) iof|by5|of8. (11) HofSfbyl^ofMofa. (12) li of 1^ of ^,- by /^ of 87^1 of 31 of ^. (13) |of2J^oflJ3of3^VbyAoflf- (14) 6^ of 31 of^^ of 34 by ^ of ^ of IJ of 19. 2. Find the continued product of (1) hhhi^^^h (2) H-, i^. i^, B?5, and li. (3) IH. 2| of 1/t. m iis, 5oW of 49, and ^. (4) f , 2|, 3j\, 5^, and 63-^- (5) ^^,-,i^,^^^,iM.andfoflf. (6) m^mhithm,^^^^^' DIYISIOK 78. Rule. Invert the divisor, i. e. take its numerator as a do- nominator and its denominator as a numerator, and proceed as in Multi- plication, Ex. Divide tt by j* Proceeding by the Rule given above. DIVISIOJ^ OF FRACTIONS. 53 2^3_^ 510 11 5 ~ 11 "^ 8 ~ 33 * Reason for the Rule. 2 2 2 If — be divided by 3, the result is — — ^ or ~ , (Art. Q6). 11 1 1 X ti oo This result is 5 times too small, or, in other words, is only one-fifth part of the required quotient, since, instead of dividing by 3, we have to divide by f , which is only one-fifth part of 3 ; and the quotient of ^ divided by f must therefore be 5 times greater than if the divisor were 3. Hence the above result ^ must be multiplied by 5 in order to give the true quotient. Therefore, the quotient = ^ x 5 = ^^^^r— = — - . oo oo So Note 1. Before applying this Rule, mixed numbers must be reduced to improper fractions, and compound fractions to simple ones, as in the following Examples : Ex. 1. Divide 4^ by 2f . ~ 4l.93_13^11 Ex. 2. Divide ^ of ^ 4 8 =¥>< 4 11 52 ~83 -m. yflofr. 4"^r -i- 3x7 . 15x7 4x8 ■ "16x1 3x7 .16x1 4x8"" " 15 x7 3x7x16 4x8x15x7 3x7x4x4 4x2x4x8x5vf 1 10' 54 ARITHMETIC. Note 2. Complex Fractions may by this Rule be reduced to simple ones. Thus, |i = i = i-"|(^"*-^^> 4:^ 6 10 • 13 1 J^ X 4 ■»' 2 Orthus, ^ = | = |_^.2, multiplying the mimeriitor and denominator of the complex fraction by the product of the denominators of the simple fractions, 20 10* # 9 80 9 1 8x3 '-^^ 2 ■ 1 2 80 2x3x10 ""20* Orthus, 3o = ^=.^Ox2xl Airain, Or thus. _9^__8_ ~C0~20' 80_80_^_80__9 4^-" f ~ I" 1 •2 _S0 2_ 3x10x2 ■" 1 ^ 9~ 8x3 20 _. = ^ = 65. 80 sj) ? P X 1 X 2 4i f fxlx2 60 20 ,.„ = ir=-8-^^^^- Ex. XXII. 1. Divide (1) 8byf. (2) fbyi (8) f by if. (4) Ifbyf. (0) flbySJ. (G) ^ by H- (7) 2#,.by4L (8) |^byfof8i. (9) 2^ by Gf of 21. (10) 8| of 8.1 of 1 by 75. (11) 3^of5|of8ilby9|ofi'jof7J. (12) 119 by |. DIVISION OF VULGAR FRACTIONS. 55 (13) g of f of CO.} of 9 by f of j of i of SJ. (14) I of i of i^ of 1/^: by ^ of ^^ of ^ of 1 A- 2. Compare the product and quotient of 2^- by 3^. o. Reduce to simple fractions the following complex fractions J (1) A. (2) |. (3) f . (4) |. .-> 13J „. 5C 13JL ^•'^ -20- • e^) If- ('^ Hf- 79, Miscellaneous Examples in Fractions worked out. Ex. 1. What number added to f 4-y% will give 2^? This question in other words is the following : " What number will [lain after ^ + xV has been subtracted from 2i?" Now " ^-{i + T\) = 21r- -i- ■A =v- -i- -A = V'- 5 "TTT — ^Stz ^F= =ff. Therefore the number required = #. N'ote. It will be remembered, that all quantities witliin a vinculum are equally affected by any sign placed before the vinculum. Thus in the above expression, — (g^ + x^) means that the sum of f and j^ has to be subtracted from 2i- ; whereas — |-+i^2 would mean that I" had to be subtracted from 2}, and then -^\ had to be added to the result. Ex. 2. What number subtracted from 14| will leave IJ for a ro- mainder ? Number required = 14| — If =.(U + l + |)-(l + l+f) = (14+V)-(2 + |) = i4-2 + (V-f) = 124. Or thus, I4|-l| = 13i-| = i£^-f ^ifL = 12f. Ex. 3. What number multiplied by If wdll produce 14| ? This question in other words is the following : '' If 14f be divided by 1^, what will the quotient be V 111 69 But iii =. X 1| ¥ 5G ARITHMETIC. 69 8 59x2 118 ■"11 ~ 11 Therefore the number required =10j\. Ex. 4. What number divided by If will produce 10-^ ? This question in other words is the following : " What is the product oflfandlOj^r The product of If and lOy^j- = ^« = 14|. Ex. 5. Reduce the expression f?i+A_j5.of ^Ui^ Vr^ioj 18^*7/ ^* to its simplest form. /Si- 2 5 „4\ ,„ =(M-^)x-I- __ 42-10 .. 7 63 Ex. 6. Simplify the expression ^ — | — a--. - _L 1- 1 -4- ii_ 126 + 90->-70 315 MISCELLANEOUS EXAMPLES IN VULGAR FRACTIONS. 57 -ii _ 13 315 _ 13x3x105 ~12^286~3x4xl3x22 Ex. 7. DividcS-l— f of^by21i + ^ + 4Xof6. '51—5 ofJL = X3 — 2 *?4 6 "^ 1 5 4- 9 _ 117-8 ~ 36 _ 109 -21j^ + 21| -21 + 21 +i + t =21 + 21 + 7- -43X 259 ~ 6 ' therefore the quotient required = ._- -^ _- 109 6^ " 36 ^ 259 ^ 109 6 " 6 X 6 ^ 259 109 ~ 1554 * Ex. 8. Simplify the expression 1 . 1 8 + i. Now, — — = 1+^ iH 58 ARITHMETIC. 1 + bV S9 39 + 4 89 43' therefore Ts^^lTI ^1^"'! _ 1- " 43' Ex. 9. Simplify I 2a+ I of ;^ -^-f U l^^. The expression n,^ .f _|) 805 4 2 L9 I f • 228 11 5 7 5 5 2) 228 4+2"l"i9-3"5hs05 11 175 2) 223 4 "^ 88 3 P 805 (the least common multiple of 4, 88, and 8, -88 x 2 ?<8) 11 X 10 X 8 + 175 X 2 X 8-2 X 88 x 2 ) 228 88x2x3 i ^ 806 _ 5 627 + 1050-152) 228 ~ I 228 ) ^ 805 1677-152 228 ~ 228 ^ 805 ^1525 ^'805 = 5. Ex. XXIII. Miscellaneous Questions and Examples on A7is. (58—70). I. 1, Define a fraction ; what is the distinction between a Vulgar and a Decimal fraction ? How many different kinds of Vulgar fractions are there ? Give an example of each kind. 91 2. Find the sum and difference of ^ of 7-2", and If divided by 2| ; and the sum of 5^, f of 8J, and f-^f . QUESTIONS AND EXAMPLES IN FRACTIONS. 59 S. Simplify 0) ]fifof5i}x{A + § + 8f}. (2) S^o!8f^^hoi9. m ^* ^^4- ^ U\ 4;^ ^4^x41-1 3, 1 4. Shew that the fraction ^ — - — ^ lies between the greatest and least + 5H-7 of tlie fractions |, f, and f . 5. The difference of two niiinhers is lo^j ; the greater number is 201^ : find the smaller mmil^er. II. 1. If tlie numerator and denominator of a fraction be both multiplied or both divided by the same number, the value of the fraction is not altered : prove this by means of an example. 2. What number subtracted from 41} leaves 19} ^ and what number multiplied by 2^^ of j% produces 81 of f 1 8. When is a fraction said to be in its lowest terms ? Reduce the fractions ffttf ^^^ Tr/yV^ to their lowest terms. 4. Simplify (I) l+rt-rl-^A- (2) sfofsjofj-jofiv (3) (A + A)^(3-J-)-(J + i)- (i) A of I of |n. 5. Divide the product of 2^ and 2| by the difference of 2| and 2i. Explain Avhy it is necessary in the addition and subtraction of fractions to reduce the fractions to a common denominator. III. 1. Shew by an example that multiplying the numerator of a fraction by any number, is the same in effect as dividing the denominator by that number, and conversely. 2. Simplify (1) 275^ + 623-V^-M08li + iof4150f (2) ff-^VV xi'^-l^- ^^ SI 9 2 4f ^^ 4I- + 3I 4-81' 8. Which is the greater, J of 4 or J of 5 ? and by how much ? 4. Divide the sum of the fractions f- and 3^3 by the product of ^j and ^ ; and reduce the result to its lowest terms. 5. What number is that, from which if you deduct f — f, and to the remainder add the quotient of 3^5 divided by 2 J, the sum will be 60 ARITHMETIC. IV. 1. Define a Vulgar fraction; an improper fraction; and the terms numerator and denominator of a fraction. Prove by means of an example the rule for the multiplication of fractions ; and multiply the sum of f of J- and 1^ by the difference of ^j and 1. 2. Reduce to their most simple forms the following expressions : (1) fx/3-x8i-fthsof(7f + t). (2) 1-3^ + ^5 -2^- (3) mn- (4) i^of(l + 5i)+^ofJ^of(7-2f)-^. 6 7 a What number added to f of (i + 1 - j^ + 1) makes 8} ? and what number divided by J of |^ of ^ w^ill give ^-^ 1 4. If I pay away J of my money, then \ of what remains, and then \ of what still remains ; what fraction of the whole will be left ? 5. Explain the method of ' comparing ' fractions. Compare the product and quotient of the sum and difference of 5 J and oj. V. 1. State the rules for multiplying and dividing one fraction by another ; and prove them by means of an example. Divide - — - by - — -^ ; and multiply the sum of ^, Ij, and f by the difference of ^ and ^, and divide the product by ^ of 1^. 2. Reduce to their simplest forms jtj 9 3_ 'L — 2\^(k — d\ (9\ 16 2 4 9 54 0) (l-l)-(il-l). (2) (8) fofil-lofif + fof^^ (4) l+^n-^nXsV- (5) 4 . l_lii 9^2 14 1 f ■^2ixli''^- ^ ofA + l-of^ iif-M 3-14— toffs* ^ 31 + 5^ 3. What is meant by the symbol f ? and ff , shall 4' 6^ make the result an integer. 4. Find the sum of the greatest and least of the fractions f , j^o, ^ and ^ ; the sum of the other two ; and the difference of these sums. QUESTIONS AND EXAMPLES IN FRACTIONS. 01 6. A man has | of an estate, he gives his son | of his share ; what portion of the estate has ho then left ? VI. -'^ V 1. State the rules for addition and subtraction of vulgar fractions; and prove them by means of an example. 2. Simplify (1) Sofi-iofA + fofllf (2) |^| + §. (3) \fxix■l^\■^{ixf+io\. (4) |i^|'^ 3. Define a proper, mixed, and compound fraction. Explain the method of reducing a compound fraction to a simi^le one. Ex. foffof^YfOfl^- 4. Shew by means of an example how a fraction is affected if the same number be added to its numerator and denominator. 20?- 41 i 5. Multiply 31- by 8j\j, and divide —^ by —^ , and find the difference between the sum and difference of these results. 6. What number added to |s-+|r will produce Sjf^? and what number divided by 2Jg- will produce y^? VII. 1. Shew from the nature of fractions that f 4- -f = |f ; that | of f = Jf ; and that f-f-f = f|. 2. Simplify (3) (3iof4J)t(2ilj)of(3^-i) (4) (i>Vof8i) + (|-§|)-(i-^)-f-(2-l). a of -^ of 3-^ 3. Simplify \ ,.1 — f^, and take the result from the sum of I Of, ■^ 01 g- 01 ^ 4. Add together i, -|, \, and i, subtract the sum from 2, multiply the result by | of |-^ of 8, and find what fraction this is of 99. 5. In a match of cricket, a side of 11 men made a certain number of runs, one obtained ith of the number, each of two others -^^ih, and each of three others -^^1^1, the rest made up between them 126; v/hich was the remainder of the score, and 4 of these last scored 5 times as many as the other. What was the whole num])er of runs, and the score of each man ? \j'Z ARITHMETIC. DECIMALS. 80. It has been stated that figures in the units' place retain their intrinsic values^ while those to the lejl of the units^ place increase tenfold at each step from the units' place; therefore, according tothe same nota- tion, as we proceed from the units' place to the right, every successive figure would decrease tenfold. We can thus represent whole numbers or integers and fractions under a uniform notation by means of figures in the units' place and on each side of it; for instance, in the number 6C73'2412, the figures on the left of the dot ' represent integers, while those on the right of the dot denote fractions. The number written at length would stand thus, 5x1000 + 0x100 + 7x10 + 3 + 3^^ + ^^^ + ^^^ + ^^^^. The dot is termed the decimal point, and all digits to the right of it are called Decimals, because they are fractions with either 10, 100, or 10 X 10, 1000, or 10 X 10 X 10, &c. as their respective denominators. 81. It may here be observed, that, when a number is multiplied by itself any number of times, the product is called a Power of that number; being called the second^ third, fourth, &c. power, according as the number is multiplied once, twice, three times, &c. by itself, that is, according as it is employed twice, three times, &c. as a factor. 82. It will be seen from what has been said, that Decimals are in fact fractions having either 10 or some power of 10, for their denomi- nators. For this reason also they are called Decimal FnAcxiONS, in contradistinction to Vulgar Fractions, which, as we have seen, are represented by a different notation, and not limited in their denominators to 10, or powers of 10. 83. From the preceding observations, it appears that First, •2845-:^ + -|-+ 4 ^ 5 10 100 1000 10000* Now the least common multiple of the denominators of the fractions is 10000 : therefore, reducing the several fractions to equivalent ones with their least common denominator, we get •2345: 2 1000 8 10 ^ 1000 ^ 100 ^ 100 100"^ 4 1000' 10 , ^10 + 5 10000 2000 + 300 + 40 + 5 10000 2345 10000' DECIMALS. U*i 3 2 4 Secondly, -00324- ^q + Joo "^ 1000 "^ TOOOO "^ 100000 (the least common multiple of the dcnommatois is 100000) 10000 _o^ M^4._L- 1^+_A__ iQ ^ "10"^ TOOOO "^ 100 "^ 1000 1000 "^ 100 10000 "^ 10 100000 300 + 20 + 4 100000 824 100000 Thirdly, 56-816 = 5 x 10 + + ^ + 3^ + to%^ (the least common multiple of the denominators is 1000) _5_xio 1000 6 1000 _5. i22 _L 1^ _?_ "~1~^1000'^1 "^1000 "^10^ 100 "^100^ 10*^1000 50000 + 6000 + 800 + 10 + 6 1000 56816 "1000* Hence, we infer that every decimal, and every number composed of integers and decimals, can be put down in the form of a vulgar fraction; with the figures comprising the decimal or those composing the integer and decimal part (the dot being in either case omitted) as a numerator, and with 1 followed by as many zeros as there are decimal places in the given number for the denominator. 84. Conversely, any fraction having 10 or any power of 10 for its de- nominator, as Yw^^ ^^^y ^^ represented in the form 56*816. ^ 56816 5x10000 + 6x1000 + 8x100 + 1x10 + 6 1000 lOOO 5x10000 6 X 1000 8x100 1x10 6 1000 '*" 1000 '*' 1000 "^ 1000 "^1000 = 6xlO + 6 + /^+iio + i4^ = 56*816 (by the notation we have assumed). 86, Again, by what has been said above, it appears that •327- ^^^ ^^^"1000 •0327: ''^ 10000 G4 APJTHMETIO. .89.70^^270 ^.827, ^^ 10000 icoo* We see that •327, •0327, and '3270 are respectively equivalent to frac- tions which have the same numerator, and the first and third of w^hicli have also the same denominator, while the denominator of the second is greater. Consequently, ^827 is equal to •3270, but ^0327 is less than either. The value of a decimal is therefore not affected by affixing cyphers to the right of it ; but its value is decreased by prefixing cyphers : which effect is exactly opposite to that which is produced by affixing and pre- fixing cyphers to integers. 86. Hence it appears that a decimal is multiplied by 10, if the decimal point be removed one place towards the right hand ; by 100, if two places ; by 1000, if three places ; and so on : and conversely, a decimal is divided by 10, if the point be removed 07ie place to the left hand ; by 100, if ttoo places ; by 1000, if three places ; and so on. Thus 5-6 xlO = |§x 10 = 56. 5-6 X 1000 = ff X 1000 = 5600. 5-6-10 = ffxJo=T'A = -56. 5-8-1000=-f^x^ioo = ioWo-'0056. 87. The advantage arising from the use of decimals consists in this; viz. that the addition, subtraction, multiplication, and division of decimal fractions are much more easily performed than those of vulgar fractions; and although all vulgar fractions cannot be reduced to finite decimals, yet we can find decimals so near their true value, that the error arising from using the decimal instead of the vulgar fraction is not perceptible. Ex. XXIV. 1. Convert the following decimals into vulgar fractions : •1; -3; -31; '311; -31111; -31111111. 2. Convert the following decimals into vulgar fractions in their lowest teiTns : •5 ; -25 ; ^35 ; •Oo ; '005 ; ^256 ; ^0256 ; -000256 ; •00008125. 8. Express as vulgar fractions in their lowest terms : •075 ; -848 ; 3-02; 3-434; 343-4; -03434 ; -050005 ; 230-409 ; 2-30409 ; 2187-2; 91800-0008; 24-000625; 8213-7169125; -00083276; 1-0000009; •OOOCOOOOl. ADDITION OF DECIMALS. 65 4» Express as decimals, rtJJ to J TU> JTTS f TIRJ 5 1000 J TOOOO i TU(J ^ lOOOOO 5 lOOOO > 9_ . 5 2 on . 00 . r? 1 4 5 . fl72S10 ■ P 7 2 R 1 . C 7 2 R 1000 1000 00 > ~ro ' TSro> 10000 ' 100000 ' loooOooooO* ~lT]U(yu^ • 6, Multiply •7 separately by 10, 100, 1000, and Ly 100000 ; •006 separately by 100, 10000, and by 10000000 ; •0431 separately by 100, and by 1000000; ] 6*201 separately by 10, 1000, and by a million ; 9"0016 by ten hundred thoiisand, and by 100. 6. Divide •51 separately by 10, 1000, and by 100000 ; •008 separately by 100, and by a million ; 5-016 separately lay 1000, and by 100000 ; 378-0186 separately by 1000, and by a million. 7. Express according to the decimal notation, five-tenths ; seven- tenths; nineteen hundredths; twenty-eight hundredths; five thousandths ; ninety-seven tenths ; one millionth ; fourteen and four-tenths ; two hun- dred and eighty, and four ten-thousandths ; seven and seven-thousandths ; one hundred and one hundred- thousandths ; one one-thousandth and one ten-millionth ; five-billionths. 6. Express the following decimals in words : •4 ; -25 ; '75 ; '745 ; -1 ; '001 ; '00001 ; 23-75 ; 2-375 ; ^2375 ; '00002375 ; I'OOOOOl ; -1000001 ; '00000001. ADDITION OF DECIMALS. 88. Rule. Place the numbers under each other, units under units, tens under tens, &c., one-tenths under one-tenths, &c. ; so that the decimals 1t)e all under each other : add as in whole numbers, and place the decimal point in the sum under the decimal point above. Ex, Add together 27-5037, '042, 342, and 2-1. Proceeding by the Rule given above, 27-5037 •042 842- 2-1 371'6457 ^ote. The same method of explanation holds for the fundamental 66 ARITHMETIC. rules of decimals^ which has been given at length in explaining the Rules for Simple Addition, Simr)le Subtraction, and the other fundamental rules in whole numbers. Reason for the aJjove process. If we convert the decimals into fractions, and add them together as suchj we obtain 27-5037 + -042 + 342 + 2-1, _ 275037 _42_ 342 21 , ~ lOOOd "^ 1000 "^ 1 ' "'' 10 ' (or reducing the fractions to a common denominator), _ 275037 10000 420 lOOOO 8420000 i"^ 10000 21000 ^ 10000 8716457 10000 = 371-6457, (Art. 84). Ex. XXV. 1. Add together: (1) -234, 14-3812, -01, 32-47, and '00075. (2) 282-15, 3-225, 21, '0001, 34-005, and '001304. (3) 14-94, -00857, 1*5, 5607'25, 530, and '0057. 2. Express in one sum : (1) -08 + 165 + 1-327 + -0003 + 2760'1 + 9. (2) 346 + -0027 + -25 + '186 + 72'505 + '0014 + '00004. (3) 6-3084 + -006 + 36-207 + '0001 + 864 + -008022. (4) 725-1201 + 84-00076 + -04 + 50-9 + 143-718. (5) 67-8125 + 27-105 + 17-5 + '000375 + 255 + 3'012o. S, Add together : (1) 2-0068, '04137, -987641, 1'0000009, 57, and 1'5 ; and prove the result. (2) -0003025, 29'99987, 143'2, 5-000025, 9000, and 3'4078,- and verify the result. (3) 21-74, '075, 103-00375, '0005495, and 4957'5 ; and verify the result. (4) Five hundred, and nine-hundredths ; three hundred and seventy-five ; twenty thousand and eighty-four, and seventy-eight hun- dred-thousandths ; eleven millions, two thousand, and two hundred and nine millionths ; eleven thousand-millionths : one billion, and one billionth. SUBTRACTION OF DECIMALS. Q7 SUBTRACTION^ OF DECIMALS. 80. Rule. Place the less number under the greater, units under units, tens under tens, &c., one-tenths under one-tenths, &c. ; suppose cyphers to be supplied if necessary in the upper line to the right of the decimal : then proceed as in Simple Subtraction of whole numbers, and place the decimal point under the decimal point above. Ex. Subtract 5-473 from 6-23. Proceeding by the Rule given above, 6-23 5-473 •757 Reason for the above process. If we convert the decimals into fractions, and subtract the one from the other as such, we obtain 623 5473 1-23 - -5-473 "ioo~ 1000 6230 1000 5473 1000 757 1000 - -757, Art. (84), Ex. XXVI. 1. Find the difference between 2-1354 and 1'0436 ; 7*835 and 2-0005 ; 15-67 and 156*7 ; 'OOl and -0009 ; *305 and '000683. 2. Find the value of (1) 213-5-1-8125. (2) '0516- -0094187. (3) 603 --6584003. (4) 17*5-13*0046. (5) -582 --09647. (6) 9*233 -'0536. 3. Take -01 from '1; 57-704 from 713-00683; 35*009876 from 56*078 ; 27*148 from 9816 ; and prove the truth of each result. 4. Required the difference between seven and seven tenths; also between seven tenths and seven millionths ; also between seventy-four + three hundred and four thousandths and one hundred and seventy-four + one hundredths ; and verify each resiilt. oy ARITHMETIC. MULTIPLICATIOISr OF DECIMALS. 90. Rule. Multiply the numbers together as if they were whole numbers, and point off in the product as many decimal places as there are decimal places in both the multiplicand and the multiplier ; if there are not figures enough, supply the deficiency by prefixing cyphers. Ex. 1. Multiply 5-34 by '21. Proceeding by the Rule given above, 5-84 •21 534 1068 11214 Now the number of decimal places in . the mr iltiplicand + the number of those in the multiplier = 2 + 2 = 4; therefore product = 1-1214. Ex. 2. Multiply 5-34 by -0021. 5-34 •0021 534 1068 11214 We must have 6 decimal places in the product; but there are only 5 figures ; and therefore we must prefix one zero, and place a point before it thus -011214. Reason for the above process. .„, „, 534 21 5-34x-21 = ^^^x- = ^-^=1-1214. 10000 fcQ4 91 Agam 5-34 x -0021 = — - x 100 10000 11214 1000000 = -011214. Ex. XXVII. 2. Multiply together : (1) 3-8 and 42 ; '38 and '42 ; 3-8 and 4*2 ; '038 and '0042. DIVISION OF DECIMALS. 69 (2) 417 and '417 ; '417 and -417 ; 71956 and 000025. (3) 2-052 and -0031 ; 4-07 and 916 ; 476 and 00026. 2. Multiply (proving the truth of the result in each case) (1) 81-4632 by -0378. (2) 27'35 by 7-70071. (3) -04375 by -0754. 3. Find the product of (1) -0046 by 7-85. (2) '00846 by -00324. (3) -314 by -0021. (4) 009 by -00846. (5) '009207 by 6 056. (6) "00948 by 29,' proving the truth of each result. 4. Find the continued product of 1, '01, '001^ and 100 ; also of *12^ 1'2, '012, and 120; and prove the truth of the results. 5. Find the value of (1) 7-6 X '071 X 2-1 X 29. (2) -007 X 700 X 760-3 X '0041 6 x 100000. BIYISIOISr OF DECIMALS. 91. First. When the number of decimal places in the dividend exceeds the number of decimal places in the divisor. Rule. Divide as in whole numbers, and mark off in the quotient a number of decimal places equal to the excess of the number of decimal places in the dividend over the number of decimal places in the divisor; if there are not figures sufficient, prefix cyphers as in Multiplication. Ex. 1. Divide 1'1214 by 5-34. Proceeding by the Rule given above, 5'34) 1'1214 (21 1068 534 534 No the number of decimal places in the dividend — the number of decimal places in the divisor = 4— 2==2,' therefore the quotient =*21. Ex. 2. Divide -011214 by 53-4. 53'4) -011214 (21 1068 634 o34 70 ARITHMETIC. Now the number of decimal places in the dividend — the number of decimal places in the divisor =6-1=5; therefore we prefix three cyphers, and the quotient is '00021; Reason for the above process. ri2M--5-34 ^ 11214 .534 ~ 10000 ■ 100 ^ 11214 100 "10000^^534 11214 100 X 534 10000 _21 1 -T^'ioo / . 11214 „, ,100 IN ^smce-^3^=21, andj-^^=~j, __21 ~100 = -21. Again, •011214^53-4 11214 534 1000000 10 11214 10 X 1000000 534 11214 10 X 534 1000000 21 1 1 ^ 100000 21 100000 = •00021. 92. Secondly. When the number of decimal places in the dividend is less than the number of decimal places in the divisor. Rule. Affix cyphers to the dividend until the number of decimal places in the dividend equals the number of decimal places in the divisor ; the quotient up to this point of the division will be a whole number ; if there be a remainder, and the division be carried on further, the figures in the quotient after this point will be decimals. DIVISION or DECIMALS. 71 Ex. Divide 1121-4 by -584. Proceeding by the Rule given above, •534) 1121-400 (2100 1068 634 534 00 Reason for tlie above process. 1121-4-f--584 _11214^ ^34^ 10 " 1000 ^ 11214 1000 " 10 "^ 534 11214 1000 534 "" 10 -21x100 = 2100. Note. In order to prevent mistakes in the proof of examples in Division of Decimals, always contrive in the process to separate 10, 100, &c. in the two fractions from the other figures, as in the above examples; and be sure never to effect the multiplication if there be tens left in the denominator; nor, if there be tens left in the numerator, to effect it until the last step of the operation. Ex. Divide 172*9 by 142 to three places of decimais- •142) 172-900000 (1217-605 142 72 ARITHMETIC. Here we must affix 5 cyphers to 172 9 ; for if we affix two according to the rule, the division up to that point will give the integral part of the quotient only, and therefore as the quotient is to be obtained to three places of decimals, we must affix three cyphers more, that is, we must affix five altogether. Reason for the above process. 172-9 ^142 _ 1729_^ 142 10 ■ 1000 _ 1 729 1000 ~ 'l42 ^ 10 ^1729 100000 142 "^ 1000 __ 172900000 1 142 ^ 1000 172900000 loi^r^A- c u Now -^— tf; — = 1-1 / 60o. . . from above ; 142 ' therefore the result ~ " 1000 " = ] 217-605. Ex. XXVIII. 1. Divide, (proving the trath of each result by Fractions) (1) 10-836 by 5-16, and 84-96818 by '881. (2) '025075 by 1*003, and '02916 by -0012. (8) -00081 by 27, and 1-77089 by 4-785. (4) 1 by -1, by -01, and by -0001. (5) 81-5 by -126, and 52 by '82. (6) 8217 by -0625, and '08217 by 6250. (7) 4-63638 by 81-34, and 15*4546 by '019. (8) -429408 by 59-64, and 2147-04 by -086. (9) 12-6 by '0012, and '065841 ])y -000475. (10) 3-012 by -0006, and 293916-669 by 541 •283. (11) 130-4 by '0004 and by 4, and 46-634205 by 4807'65. (12) 1-69 by 1-3, by -13, by 18, and also by -013. (18) -00281 by 1405, by 1405, and by -001405. (14) 72-36 by 36 and by '0036, and -003 by 1'6. (15) G725402'3544 by 7089, and by '7089. (16) 10363284-75 by 89G'25, and '09844 by '0046, DIVISION OF DECIMALS. 73 (17) 816 hy 000-1, and -0019610652875 by 2*38645. 08) 18368830 5 by 2315, by 231-5, and by -2315. (19) -00005 by 2'5, by 25, and by -0000025. (20) 684-1197 by 1200'21, and also by -0120021. 2. Divide to four places of decimals each of the following, and prove the truth of the results by Fractions : (1) 32-5 by 8-7 ; '02 by 1-7 ; 1 by -013. (2) -009384 by '0063 ; 51846-784 by 1-02. (3) 7380-964 by '023 ; 65 by 3-42 ; 25 by 19. (4) 176432 76 by -01257 ; 74571345 by 6535496-2. (5) 37-24 by 2-9 ; '0719 by 27 53. 3. Find the quotient (verifying each result) of (1) -0029202 by 157, and by 1-57. (2) 6005 by 1953125; of 5005 by 195-3125; of '05005 by -0001953125. (3) (7J of 1 + li) by -0005 ; of 81-008 by ff f of li of ^^ ; '7575 by 16|. 93. Certain Vulgar Fractions can be expressed accurately as Decimals. Rule. Reduce the fraction to its lowest terms ; then place a dot after the numerator and affix cyphers for decimals ; divide by the de- nominator, as in division of decimals, and tho quotient will be the decimal required. Ex. 1. Convert § into a decimal. 5 I 3-0 •6 There is one decimal place in the dividend and none in the divisor ; therefore thei-e is one decimal place in the quotient. Note. In reducing any such fraction as f^ or -^ to a decimal, we may proceed in the same way as if we were reducing % ; taking care however in the result to move the decimal point one place further to the left for each cypher cut off. Thug 5 % 3 _ 50" •06, •000, 74 ARITHMETIC. for in fact, we divide by 5, and then by 10, 100, &c., according as the divisor is 50, 500, &c. 5 Ex. 2. Reduce tt. to a decimal. IG 16) 5-0000 (-3125 48 20 16 40 82 80 80 or thus ^ = •3125 8 3 Ex. 8. Convert — and ^^j^ into decimals. Now 512 = 8x64 = 8x8x8 8-000 •875000 •040875000 •005859375 or — is equivalent to -005859375, and ^^^ is equivalent to -00005859875. Ex. 4. Convert | + 8^ + 2^ + 6 JJg- into a decimal. t + Si + 2-,% + 6A = ll + f + i + ^ + T¥5. 3-0 8 1-000 •125 ' 11 5 2-20 5 -440 4 I 9-00 2-25 9 —'OOK therefore •088 | = .6, i = -125, ^o = '225, i^ = '0S8; therefore the whole expression = 11 + -6 + -125 + -225 + '088 = 12-088. VULGAR FRACTIONS EXPRESSED AS DECIMALS. 75 Ex. XXIX. 1 . Reduce to decimals : \^J 4 ' 4= > S' 25 > TE > 20' /■9A C>C> • fi4^ • S70 . 170 . _1 K^^J TlTS > 125? 200^ 125 > 16 '4: > Si 25> Tff' 2 0* 1_ 0" 588 (3) 6ii;^;^^;^;15f|f|5. 2. Reduce to decimals : (1) 8fof-,-t>> (2) h + l + ^ + ^. (8) ^x-0064. 475 113. (4) f + -061, (5) J + i-J- (6) ^»f^*- (7) ^|?of||of§2. (8) Sjf^ + -75ofJof7i. •7^ 247 17 11 (9) 8.V + A% + 81tM^ + ^. (10) y + i^5j^2 + _|+2003^ + ^. Note. 10 is sometimes called thefrst power of 10, 10 X 10 second power of 10, 10 X 10x10 " third power of 10, 10 X 10 X 10 X 10 X 10 fflh power of 10, and so on ; similarly of other numbers. 94. We have seen that, in order to convert a vulgar fraction into a decimal, after reducing the fraction to its lowest terms and affixing c^'phers to the numerator, we have in fact to divide 10, or some multiple of 10 or of its powers, by the denominator of the fraction : now 10 = 2 x 5, and these are the only factors into W' hich 10 can be broken up ; therefore, when the fraction is in its lowest terms, if the denominator be not composed solely of the factors 2 and 5, or one of them, or of powers of 2 and 5, or one of them, then the division of the numerator by the denominator will never terminate. Decimals of this kind, that is, -which never terminate, are called indeterminate decimals, and they are also called Circulating, Repeating, or Recurring Decimals, from the fact that when a decimal does not terminate, the same figures must come round again, or recur, or be repeated : for since we always affix the same figure to the dividend, namely a cypher, whenever any former remainder recurs, the quotient will also recur. Now when we divide by any number, the remainder must always be less than that number, and therefore some remainder must recur before we have obtained a number of remainders equal to the number of units in the divisor. 95. Pure Circulating Decimals are those which recur from tho beginning : thus -3333..., •272727..-, are pure circulating decimals. 76 ARITHMETIC. Mixed Circulating Decimals are those which do not begin to recur, till after a certain number of iigiires. Thus -128888 , -0118636 , are mixed circulating decimals. The circulating part, or the part which is repeated, is called the Period or Repetend. Pure and mixed circulating decimals are generally written down only to the end of the first period, a dot being placed over the first and last figures of that period. Iriius -3 represents the pure circulating decimal -383... -86 . -3636... •639 -689689.. •188 mixed -1888... -01136 -0118686... 96. Pure Circulating Decimals may be converted into their equivalent Vulgar Fractions by the following Rule. Rule. Make the period or repetend the numerator of the fraction, and for the denominator put down as many nines as there are figures in the period or repetend ; this fraction, reduced to its lowest terms, will be the fraction required. Note. The fraction is only reduced to its lowest terms for the sake of exhibiting it in its simplest form. It is not of course actually necessary so to reduce it. Exs. Reduce the following pure circulating decimals, '3, '2f, •857142, to their respective equivalent vulgar fractions. Proceeding by the Rule given above, "-9-3* •27 = ?^ = ^. 99 11 ' • _ 857142 _ 9 5288 ■^^'^^-"999999 "111111 _ 6 X 158 73 ~ 7 X 15873 ^6 7* '^ The truth of these results will appear from the following considerations. Let the circulating decimal •3333... be represented by a symbol oc; then a; = 'SQ33... CIRCULATING DECIMALS. 77 therefore 10 times ^ = 10 times -3333.. . = 3-3333... (Art. 86). Now 10 times x, diminished by 1 time x, will leave 9 times (3|-— f j of y as decimals. (2) Simplify 4-255 X -032 '00016 * V. 2. (h+HiH+^)^ii+iH+^). 6—2 84 ARITHMETIC. a (frof85i-3i) + (2-5625 + 7i). 4. -SOS-f-l-TSx •i36---072. (3) State at leiigth the advantages which decimals possess over vulgar fractions ; what disadvantages have they ? Shew whether ^f or |^|| is nearer to the number 8*14159. (4) Find the value of 1 + :r + _ — -+ - — ^--^ -i-&c., to 7 places of decimals ; and also of lO^^^V 10^^ 1x2^^ 10' '^1x2x8'^ 107 expressing it (1) as a decimal, and (2) as a fraction. (5) Find the Earth's equatoreal diameter m miles, supposing the Sun's diameter, which is 111'454 times as great as the equatoreal diameter of the Earth, to be 883345 miles. (6) In what sense is a vulgar fraction said to be the value of a recur- ring decimal ? Explain how a sufficient degree of accuracy may be ob- tained in the addition and subtraction of circulating decimals to any given number of decimal places, without converting the decimals into fractions. Ex. Find the sum of '125, 4-I63*, and 9'457, correct to 5 places of decimals. VI. (1) Prove the Rule for Multiplication of decimals by means of the example 404*04 multiplied by •030303. Multiply -345 by -^; and divide -04813489963 by -6593, and by -006593. (2) Explain the meaning of 7^, and 7^ ; and find what vulgar fraction is equivalent to the sum of 20*5 and 2*05 divided by the difference. /o\ -D ;i i *T, • 1 w 123-48 , 36-595 (3) Reduce to their lowest terms , and ^ . rA\ ct, ^1 •375x^375 --025x025 2 , ^, , (4) Shew that ~~ — ^-^ = - , and that ^'oio — VZo 5 3,1 T "" 7 + i;- 3-14159 nearly. 16 Reduce •1293131 to its equivalent vulgar fraction. (5) What decimal added to the sum of 1^, |, and ^ will make the sum total equal to 3 ? (6) The quotient being 2J^ and the divisor '16, find the dividend. CONCRETE NUMBERS. TABLES. 85 CONCKETE NUMBERS. TABLES. 100. Our operations hitlierto have been carried on with regard only to abstract numbers, or Concrete numbers of one denomination. It is evident that if concrete numbers were all of one denomination ; if, for instance, shillings were the only units of money, yards of length, years of time, and so on, such numbers would be subject to the common rules for abstract numbers. Again, if the concrete numbers were of ditfercnt denominations, and those denominations differed from each other by 10 or multiples of 10, then all operations with such concrete numbers could be carried on by the rules which have been given for Decimals. But generally with concrete numbers such a relation does not hold between the different denominations, and therefore it is necessary to commit to memory tables, which connect the different units of money together, the different units of length together, the different units of time together,, and so on. We shall now put do^^^l some of the most useful of these tables, with a few brief remarks on each. TABLE OF MONEV. 2 Fartliings make 1 Half-penny. 2 Half-pence 1 Penny. 12 Pence 1 Shilling. 20 Shillings 1 Pound. Pounds, shillings, pence, and farthings were formerly denoted by £, s, d, and q respectively, these letters being the first letters of the Latin words libra, solidus, denarius, and quadrans, the Latin names of certain Roman coins or sums of money. £, s, d are still the abbreviated forms for pounds, shillings, and pence respectively; but i annexed to pence denotes 1 farthing, ^ denotes a half-penny, f denotes three far- things; shewing that one farthing, two farthings, and three farthings are respectively ith, f-ths or i a\v\ fths of the concrete unit, one penny. 8t) AEITHMETIC. The following coins are at present in common use in England i Copper Coiris. A Farthing, the coin of least value. A Half-penny = 2 Farthings. A Penny = 4 Farthings. Silver Coins, Threepenny-piece = 8 Pence. Fourpenny-piece =4 Pence. A Sixpence = 6 Pence. A Shilling = 12 Pence. A Florin = 2 Shillings. A Plalf-Crown .... = 2 Shillings and 6 Pence. A Crown ~5 Shillings Gold Coins. A Half-SoYereign = 10 Shillings. A Sovereign =20 Shillings. The following coins have been in use at various periods in England, but with the exception of the first two, which are used under different names, they are now obsolete : Silver Coins. A Groat = 4 Pence. A Tester = Pence. Gold Coins. £ s. d. A Noble = G 8 An Angel =0 10 A. Half-Guinea... =0 10 6 AMarkorMerk = 13 4 A Guinea = 1 1 ACarolus =1 8 A Jacobus = 1 5 AMoidore ^1 7 Note. The office at which coin is made and stamped, so as to pass or become current for legal money, is called the Mint. The standard of gold coin in this kingdom is 22 parts of pure gold and 2 parts of copper, melted together. From a pound Troy of standard TABLES — MONEY. 87 gold there are coined at the Mint 46|--j sovereigns, or X46. 14.9. 6d. : there- fore the Mint price of gold is -^ of £40. 14?. 6d. or £3, 17-?- lO^rf. per ounce standard, (12 ounces Troy = l pound Troy). The standara of silver coin is 87 parts of pure silver and 3 parts of copper. From a pound Troy of standard silver are coined QQ shillings. Therefore the Mint price of silver is 5s. 6d. per ounce standard. In the copper coinage, 24 pence are coined from 1 pound Avoirdupois of copper. Therefore 1 penny should Aveigh o'ith of a pound Avoirdupois. The copper coinage is not, according to the present law, a legal tender for more than 12c?.; nor is the silver coinage for more than 40^. j the gold coinage being the standard of this country. MEASTJEES OF WEIGHT. TABLE OF TROY WEIGHT. 101. This table derives its name probably from Troyes in France, the first city in Europe where it was adopted. It seems to have been brought thither from Egypt. It has also been derived from Troy-novant, the monkish name for London. It is used in weighing gold, silver, diamonds, and other articles of a costly nature ; also in determining specific gravities ; and generally m philosophical investigations. The different units are grains (written grs.), pennyweights (dwts.), ounces (oz.), and pounds (lbs. or lbs.), and they are connected thus ; 24 Grains make 1 Pennyweight ... 1 dwt. 20 Pennyweights 1 Ounce 1 oz. 12 Ounces 1 Pound 1 lb. or lb. ]Vote 1. As the origin of weights, a grain of wheat was taken from the middle of the ear, and being well dried, was used as a weight, and called 'a grain.' Note 2. Diamonds and other precious stones are weighed by ' Carats,' each carat weighing about 3^ grains. The term ' carat ' applied to gold has a relative meaning only ; any quantity of pure gold, or of gold alloyed with some other metal, being supposed to be divided into 24 ecj^ual parts (carats) ; if the gold be pure, it is said to be 24 carats fine; if 22 parts be pure gold and 2 parts alloy, it is said to be 22 carats fine. Standard gold is 22 carats fine : jewellers' gold is 18 carats ftne, 88 ARITHMETIC. TABLE OF APOTHECARIES' WEmHT. 102. Apothecaries* weight only differs from Tl'oy weight in the subdivisions of the pound, which is tlie same in both. This table is used in mixing medicines. The different units arc grains (grs.), scruples (^)j drams (3), ounces (g)^ pounds (lbs. or lbs.), and they are connected thus : 20 Grains... make 1 Scruple ... 1 sc. or 1 3. o Scruples ....... 1 Dram 1 dr. or I 3. 8 Drams 1 Ounce .... 1 oz. or 1 %. o 12 Ounces.. 1 Pound .... 1 lb, or ib. TABLE OF AVOIRDUPOIS WEIGHT. 103. Avoirdupois weiglit derives its name from Avoirs (goods or chattels, and Poids (weight). It is used in weighing all heavy articles, which are coarse and drossy, or subject to waste, as butter, meat, and the like, and all objects of commerce, with the exception of medicines, gold, silver, and some precious stones. The different units are drams (drs.), ounces (oz.), pounds (lbs.), quarters (qrs.), hundredweights (cwts.), tons (tons), and they are connected thus : 16 Drams make 1 Ounce 1 oz. 16 Ounces 1 Pound 1 Ib. 28 Pounds 1 Quarter 1 qr. 4 Quarters 1 Hundredweight... 1 cwt. 20 Hundredweights 1 Ton 1 Ton. In general, 1 Stone (1 st.) = 14lbs. Avoirdupois, but for butchers* meat or fish, 1 Stone = 8 lbs. ; 1 Firkin of Butter = 56 lbs.; 1 Fodder of Lead = 19i^ cwt. ; 1 Great Pound of Silk = 24 ounces; 1 Pack of Wool = 240 pounds. 1 lb. Avoirdupois weighs 7000 grains Troy ; 1 lb. Troy weighs 5760 grains Troy j therefore 1 lb. Avoirdupois = ^j^ of 1 lb. Troy = 1^ of lib. Troy = -H-f-ofllb. Troy = 14 oz. 11 dwt. 16 grs. Troy = 1 lb- 2 oz. 11 dwt. 16 grs. Troy. TABLES — LENGTH, "89 MEASURES OP LENGTH, TABLE OF LINEAL MEASURE. 104. In this measure, wliicli is used to measure distances, lengths breadths, heights, depths, and the like, of places or things : 3 Barley-corns (in length) make 1 Inch, which is written 1 in. 12 Inches 1 Poot, ..1ft. 5 Feet . 1 Yard, 1yd. 6 Feet 1 Fathom • 1 fth. Bh Yards 1 Rod, Pole, or Perch ... 1 po. 40 Poles 1 Furlong-, 1 fur. 8 Furlongs 1 Mile, 1 m. 5 Miles 1 League, 1 lea. 69^ Miles 1 Degree 1 deg. or 1". A^ote, A gTain of Barley, or a Barley-corn, is supposed to have been the origmal element of Lineal Measure. The following measurements may be added, as useful in certain cases : 4 Inches make 1 Hand (used in measuring horses), 22 Yards make 1 Chain ^ . . • i i , ^ T . 1 1 -. ^1 . f used ni measurmg land, 100 Links make 1 Cham J ^ a PalmOS inches, a Span = 9 inches, a Cubit =^18 inches, a Pace = 5 feet, 1 Geographical Mile = g'(j"' of a degree, a Line = ^*^ of an inch. TABLE OF CLOTH MEASURE. 105. In this measure, which is used by linen and woollen drapers : 21 Inches make 1 Nail. 4 Nails 1 Quarter ... 1 qr. 4 Quarters ... 1 Yard 1 yd. 5 Quarters ... 1 English Ell. 6 Quarters ... 1 French Ell. 3 Quarters ... 1 Flemish Ell. 90 ARITHMETIC, MEASURES OF SURFAOK TABLE OF SQUARE MEASURE. 106. This measure is used to measure all kinds of superficies, such as land, paving, flooring, in fact everything in which length and breadth are to be taken into account. Def. a square is a four-sided figure, whose sides are equal, each side being perpendicular to the adjacent sides. A square inch is a square, each of whose sides is an inch in length ; a square yard is a square, each of whose sides is d yard in length. 144 Square Indies make 1 Square Foot...l sq. ft. or 1 ft. Square Feet 1 Square Yard... 1 sq. yd. or 1 yd. 80^ Square Yards 1 Square Pole...l sq. po. or 1 po, 40 Square Poles 1 Square Rood 1 ro. 4 Roods 1 Acre 1 ac. 25000 Square Links ~1 Rood, 100000 = 1 Acre. 10 Square Chains = 1 Acre. Note. This table is formed from the table for liijeal measure, by multiplying each lineal dimension by itself. *rhe truth of the above table will appear from the following considera^ iions. Suppose AB and J C to be lineal yards placed perpendicular to each other. Then by definition ABCD is a square yard. If AE, ^ ^' ^ ^ EF, FB, AG, GH, HC-=1 lineal foot each, it appears ^ from the figure that there are 9 squares in the square yard, and that each square is 1 square foot. The same explanation holds good of the other di- mensions. The following measurements may be added : A Rod of Brickwork = 2721 Square Feet. (The work is supposed to be 14: in., or rather more than a hrick-and-a- half, thick) 4 7 2 5 8 3 6 TABLES — MEASURES OF SOLIDITY. 91 A Square of Flooring .... = 100 Square Feet. A Yard of Land..., =30 Acres. A Hide of Land = 100 Acres. MEASURES OE SOLIDITY. TABLE OF SOLID OR CUBIC MEASURE. 107. This measure is used to measure all kinds of solids, or figures which consist of three dimensions, length, breadth, and depth or thick- ness. Def. a cube is a solid figure contained by six equal squares ; for instance, a die is a cube. A cubic inch is a cube whose side is a square inch. A cubic yard square yard. 12x12 X 12 or 1728 cubic inches make 1 cubic foot. 8x3x 8x)r 27 cubic feet 1 cubic yard. Note. This table is formed from the table for lineal measure by multiplying each lineal dimension by itself twice. The truth of the above table ivill appeal' from the following considera^ tions. li AB, AC, and AD be perpendiculaf to each other, and each of them a lineal yard in length, then the figure DE is a cubic yard. Suppose JDH a lineal foot, and HKLM a plane drawn parallel to side DC, By last table there arc 9 square feet in side DC. There will tlierefore be 9 cubic feet in the solid figure DL. Similarly if another lineal foot HN j^ were taken, and a plane NO were drawn parallel to HL, there would be 9 cubic feet contained in the solid figure HO. Similarly, there would be 9 cubic feet in the solid figure NE. Therefore, there are 27 cubic feet in the solid figure DE, or in 1 cubic yard. The following measurements may be added : A Load of rough Timber =40 cubic feet. A Load of squared Timber = 50 cubic feet. A Ton of Shipping =42 cubic feet. J) H Tf \ 1 ^ K \ 1 \ ; \ K •' \ \ \ 3 ; \ ; y.-M: B Kj^ \ \ E 92 ARITHMETIC. MEASURES OF CAPACITY. TABLE OF WINE MEASURE. 108. In this measure, by which wines and all liquids^ with tho exception of malt liquors and water, are measured : 4 Gills make 1 Pint 1 pt. 2 Pinfs 1 Quart 1 qt. 4 Quarts 1 Gallon 1 gal. 10 Gallons.... 1 Anker 1 ank. 18 Gallons.... 1 Runlet ....... 1 run. 42 Gallons .... 1 Tierce 1 tier. 2 Tierces. ... 1 Puncheon .... 1 pun. 63 Gallons.... 1 Hogshead ... 1 hhd. 2 Hogsheads 1 Pipe 1 pipe. 2 Pipes 1 Tun 1 tun. TABLE OF ALE AND BEER MEASURE. 109. In this measure, by which all malt liquors and water arc measured : 2 Pints make 1 Quart .... 1 qt. 4 Quarts 1 Gallon .... 1 gal. 9 Gallons 1 Firkin .... 1 fir. 18 Gallons 1 Kilderkin 1 kil. 36 Gallons. 1 Barrel ... 1 bar. 1^ Barrels or 54 Gallons... 1 Hogshead 1 hhd. 2 Hogsheads 1 Butt 1 butt. 2Butts 1 Tun 1 tun. TABLE OF CORN OR DRY MEASURE. 110. In this measure, by which all dry commodities, as corn, and the like, which are not usually heaped above the measure, are measured : 2 Quarts make 1 Pottle 1 pot. 2 Pottles 1 Gallon 1 gal. 2 Gallons ,.... 1 Peck 1 pk. 4 Pecks 1 Bushel 1 bus. 2 Bushels 1 Strike 1 str. 4 Bushels 1 Coomb 1 coomb. 2 Coombs or 8 Bushels 1 Quarter ... Iqr. 5 Quarters 1 Load 1 load. 2 Loads or 10 Quarters...! Last 1 last. TABLES MEASURES OF CAPACITY. 98 TABLE OF COAL MEASURE, llL In this measure, which is not much used noWj as coals are sold by weight : 4 Pecks make 1 Bushel. 3 Bushels 1 Sack. 86 Bushels 1 Chaldron. MEASURES OF NUMBER. TABLE OF NUMBER. 112. 12 Units ...=.. " make 1 Dozen. 12 Dozen 1 Gross. 20 Units 1 Score. 120 Units 1 Long hundred. 24 Sheets of Paper 1 Quire. 20 Quires 1 Ream. 10 Reams 1 Bale. MEASURES OE TIME. TABLE OF TIME. IISJ, 1 Second is written thus 1". 60 Seconds make 1 Minute .....o. P. 60 Minutes ...... 1 Hour....... 1 hr. 24 Hours 1 Day ,.. 1 day. 7 Days... 1 Week 1 wk. A year is divided into 12 months, called Calendar Months, the num- ber of days in each of which are easily remembered by means of the following lines : Thirty days hath September, April, June, and November : February has twenty-eight alone, And all the rest have thirty-one : But leap-year coming once in four, February then has one day mure. A day, or rather a mean solar day, which is divided into 24 equal portions, called mean solar hours, is the standard unit for the measure- ment of time, and it is the mean or average time which elapses betw^een two successive transits of the Sun across the meridian of any place. 94 ARITHMETIO. , The time between the Sun's leaving a certain point in the Ecliptic and its return to that point consists of 865*242218 mean solar days, or SG5 days, 5 hours, 48 minutes, 47^ seconds, very nearly, and is called a solar year. Therefore the civil or common year, which contains 365 days, is about ^th of a day less than the solar year ; and this error would of course in time be very considerable, and cause great confusion. Julius Caesar, in order to correct this error, enacted that every 4th year should consist of 366 days ; this was called Leap or Bissextile year. In that year February had 29 days, the extra day being called ^the Intercalary' day. But the solar year contains 365'242218 days, and the Julian year contains SG5-25 or 365L days. Now S65-25 ~ 865-242218 = -007782. Therefore in one year, taken according to the Julian calculation, the Sun would have returned to the same place in the Ecliptic '007782 of a day before the end of the Julian year. Therefore in 400 years the Sun would have come to the same place in the Ecliptic -007782 x 400 or 8-1128 days before the end of the Julian year; and in 1257 years would have come to the same place, -007782 x 1257 or 9"7819, or about 10 days before the end of the Julian year. Accord- ingly, the vernal equinox which, in the year 825 at the council of Nice, fell on the 21st of March, in the year 1582 (that is, 1257 years later) happened on the 11th of March ; therefore Pope Gregory caused 10 days to be omitted in that year, making the 15th of October immediately succeed the 4th, so that in the next year the vernal equinox again fell on the 21st of March; and to prevent the recurrence of the error, ordered that for the future in every 400 years, 8 of the leap years should be omitted, viz. those which complete a century, the numbers expressing which century, are not divisible by 4; thus 1600 and 2000 are leap years, because 16 and 20 are exactly divisible by 4 ; but 1700, 1800, and 1900 are not leap years, because 17, 18, and 19 are not exactly divisible by 4. This Gregorian style, which is called the new style, was adopted in England on the 2nd of September 1752, when the error amounted to 11 days. The Julian calculation is called the old style : thus Old Michaelmas and Old Christmas take place 12 days after New Michaelmas and New Christmas. In Russia, they still calculate according to the old style, but in the other countries of Europe the new style is used. Sir Harris Nicolas in TABLES — IMPERIAL STANDARD MEASURE. 95 his Chronology gives the dates at which the new style was adopted in different countries. Of course it was almost immediately adopted by most of the Koraan Catholic courts of Europe. TABLE OF ANGULAR MEASURE. 114. 1 Second is written 1 sec. or V\ 60 Seconds make 1 Minute 1 min. or 1'. 60 Minutes 1 Degree 1 deg. or 1**. 90 Degrees 1 Right Angle... 1 rt. ang. or 90°. The circumference of eveiy circle is considered to be divided into S60 equal parts, each of which is often called a degree, as it subtends an angle of 1" at the centre of the circle. 115. An Act of Parliament "for Ascertaining and Establishino Uniformity op Weights and I\1easures/' in this kingdom, came into operation on the first of January, 1826. It is thereby enacted. First; that the b7'ass Standard Yard of 1760, then in custody of the Clerk of the House of Commons, shall be the Imperial Standard Yard, (the brass being at the temperature of 62° by Fahrenheit's thermometer) ; and that this Imperial Standard Yard shall be the unit or only standard measure of extension, wherefrom or whereby all other measures of ex- tension whatsoever, whether the same be lineal, superficial, or solid, shall be divided, computed, and ascertained ; and that the thirty-sixth part of this yard shall be an Inch. Now the length of a Pendulum vibrating seconds in the latitude of London, in a vacuum, and at the level of the sea, is found to be 89"1393 such inches, i, e. 39 such inches and 1393 ten-thousandths of another such inch. This affords the means of recovering tlie Imperial Standard Yard should it be lost. In fact, the brass Standard Yard of 1760 was de- stroyed or rendered useless by the fire at the House of Commons in 1834. Secondly ; That the brass weight of one Pound Troy of the year 1758, then in the custody of the same officer, shall continue the unit or Stand- ard Measure of Weight, from which all other weights shall be derived^ computed and ascertained; that 5760 grains shall be contained in the Imperial Standard Troy Pound, and 7C00 such grains in the Avoirdupois )?ound. 96 ARITHMETIC. Now the weight of a cubic inch of distilled water is 252*458 grains Troy, the barometer being at 30 inches and the thermometer at 62°. This affords the means of recovering the Imperial Standard Pound should it be lost. In fact, the brass weight of 1758 was destroyed or lost at the above-mentioned fire. Thirdly; That the Standard Measure of Capacity for Liquids and Dry Goods shall be 'Hhe Imperial Standard Gallon," containing 10 Pounds Avoirdupois weight of distilled water, weighed in air at a tem- perature of 62° Fahrenheit's thermometer, and the barometer being at 30 inches. Now this weight fills 277'274 cubic inches, therefore the Imperial Standard Gallon contains 277*274 cubic inches. The Imperial Bushel, consisting of eight gallons, will consequently be 2218192 cubic inches. EEDUCTION. 116. Reduction is the method of expressing numbers of a superior denomination in units of a lower denomination, and conversely. Thus £1 is of the same value as 240c?., and £21 as 5040c/., and conversely ; and the process, by which we ascertain this to be so, is termed Re- duction. First. To eiXpress a number of a higher denomination in units of a lower denomination. Rule. *' Multiply the number of the highest denomination in the proposed quantity by the number of units of the next lower denomina- tion contained in one unit of the highest, and to the product add the number of that lower denommation, if there be any in the proposed quantity ; repeat this process for each succeeding denomination till the required one is arrived at." Ex. 1. How many pence are there in £23. 15^. ? Proceeding by the Rule given above, £23 .15*. 20 460 + 15 or 4755. 12 5700d. or £28. 15*. = 5700c?. REDUCTION. 97 Iteasonfor the process. There are 20 shillings in £1. Therefore there are (23 x 20)^. or 4G0*. in X'28, and so there are 4G05. + 15*., or 475*. in £23. lo*. Again, since there are 12 pence in \s. ; therefore there are (47o x 12)(/., or 5700c?. in 475^. i.e. in £23. 15^. Ex. 2. Reduce 2 tons, 7 cwt., 8 qrs,, 24 lbs. into lbs. tons cwt. qrs. lbs. 2 . 7 . 3 . 24 20 40 + 7 or 47 cwt. 4 188 + 3 or 101 qrs. 28 1528 382 5348 + 24 or 5372 lbs. Ex, 3. Reduce 27 acres, 1 rood, 32 poles, into poles. acres roods poles 27 . 1 . 82 4 108 + 1 = 109 ro. 40 4360 + 32 = 4392 poles. Ex. 4. Reduce 73 days, 21 hours, 10 minutes, 9 seconds, to seconds. days hrs. min. sec. 73 . 21 . 10 . 9 _2£ 292 146 1752 + 21:^ 1778 hrs. 60 106380 + 10 i= 106390 min. 60^ 6383400 + 9 6383409 sec 98 ARITHMETIC. Ex. 5. How many inches are there in 106 miles, 6 furlongs, 25 perches, and 2^ yards ? miles fur. per. yds. 106 o 6 . 25 . 2| 8 848 + 6 = 854 fur. 40 34160 + 25 per. = 34185 5| 170925 I7092ir 1880171 + 2i yds, =^188020 36 1128120 564000 6768720 in. Secondly. To express a number of inferior denomination in units of a higher denomination. Rule. " Divide the given number by the number of units which connect that denomination with the next higher, and the remainder, if any, will be the number of surplus units of the lower denomination. Carry on this process, till you arrive at the denomination required." Ex. 1. How many pounds and shillings are there in 5700 pence ? Proceeding by the Rule given above> 12 5700 2,0 47,5 £28. Us. In dividing 475 by 20 we cut off the and 5 by Art. (48). Reason for the above process. Since 12 pence = 1 shilling ; therefore in any given number of pence, for every 12 pence there is 1 shilling, so that in 5700d or (]2x475)fZ there are 475^. Again, since 20,?. — £1 ; therefore in any given number of shillings, for every 20 shillings there is £1. REDUCTION. 99 Hence, in 475^., or (20 x 23 + 15)5. there are £23, and 15*. over. Note. Since each of the above Rules is the converse of the other, the accuracy of any result obtained by either of them may be tested by working the result back again by the other rule. Ex. 2. In 272668 inches how many miles &c. are there ? Verify the result. In this Example it will be convenient to bring the inches to half-yards and the half-yards to poles. In a lialf-yard there are 18 or 3 x 6 inches, and in a pole there are 5^ yards or eleven half-yards. 8 18 G 272668-1 ) 90889 -] ) 11 15148-1 half-y 4,0 137,7-17 po. 8 34-2 fur. 4 therefore the answer is 4 miles, 2 fur., 17 po., 22 in. miles fur. poles in. Proof 4 . 2 . 17 . 22 miles fur. poles 4 . 2 . 17 . ^ 84 furlongs 40 1360 + 17 = 1377 poles 1377 11 15147 half-yard^ 18 121176 15147 22 272668 inches. Ex. 8. Reduce 5813456 pounds to tons, and prove the con-ectness of the result. V-~2 100 ARITHMETIC. 28 I' 4 2,0 lbs. 5813456-0 12 lbs. 1458364-8 ■8 qrs 207628- 5190,5-5 cwt. 2595 therefore the answer is 2595 tons, 5 cwt., 8 qrs., 12 lbs. tons cwt. qrs. lbs. Proof 2595 . 5 . 3 . 12 20 51905 cwt. 4 207623 qrs. 28 1660984 415246 12 5813456 lbs. Ex. 4. How many grains of gold are contained in 9 lbs., 11 oz. 13 dwts., 20 grs. 1 Prove the result. lbs. oz. dwts. Rrs. 9 . 11 . 13 . 20 108 + 11 = 119 oz. in 9 lbs., 11 oz. 20 2380 + 13 or 2893 dwts. in 9 lbs., 11 oz., 18 dwts. 24 9572 4786 or 57432 + 20 57452 grs. in 9 lbs., 11 oz., 13 dwts., 20 grs. Proof 24 4 6 2,0 12 57452-0 14368-5 20 grs. 239,8 119-13 dwts. 9-11 oz. therefore in 57452 grs., there are 9 lbs., 11 oz., 13 dwts., 20 grs. REDUCTION, 101 Ex. 5. Reduce 49 acres, 28 poles, 10 yards, feet, 112 inches, to inches. Prove the result. Proof ac. po. yds. 49 . 28 . 10 4 196 ro. 40 7840 + 28 = 7868 poles 80^ 236040 1967 112 2380074-10 := 238017 yards 9 2142153 + 8 = 2142161 feet 144 8568644 8568644 2142161 808471184 + 112 = 308471296 inches 144 12 sq. in. 808471296 12 25705941 9 2142161 h ■4 9 •8 sq.ft. 12 sq. In. 238017 Now, since 301 or ^-^ sq. yds. = 1 sq. po., we multiply by 4, which reduces the sq. yds. into quarters of sq. yds., and then divide that result by 121, or 11 x 11, which brings it into sq. poles. 238017 4 - 7 ) 40 quarters of sq. yds. -3 for 10 sq. yds. -28 sq. po. 49 121 '11 11 952068 86551 - 4,0 786,8- 4 196 102 ARITHMETIC. Therefore in 308471296 sc[. in., there are 49 ac, 28 sq. po., 10 sq. yds., 8 sq. ft., 112 sq. in. Ex. 6. How many half-guineas are there in 537 half-crowns ? Here hoth the Rules are requisite. By Rule 1, 537 half-crowns = (537 X 5) sixpences = 2685 sixpences. Next, to find how many half-guineas there are in 2685 sixpences. By Rule 2, 8 21 ■ 7 2685-0) — > 18 sixpences ; 15-6) 895-6 127 therefore in 537 half-crowns, there are 127 half-guineas and 18 sixpences. Ex. XXXIII. (1) Reduce (verifying each result) ; 1. £57 to pence ; and 618 guineas to farthings. 2. £15. 12*. to pence ; and 5000 guineas to pence. 8. 85. A^d. to half-pence ; and £1. Os. Sfc?. to farthings. 4. £83. 15*. Gld. to farthings ; and £393. 0*. IIM to half-pence. 5. 788 half-crowns to farthings; and 570 crowns to fourpenny pieces. 6. 2673 half-guineas to farthings j and 22| guineas to sixpences. (2) Find the number of pounds in 5673542 farthings, and prove the truth of the result. (3) How many half-crowns, how many sixpences, and how many fourpences, are there in 25 pounds ? (4) In 6300 fourpences, how many half-crowns are there, and how many half-guineas 1 (5) In 851 seven-shilling-pieces, how many half-guineaa are there, and how many moidores 1 (6) Reduce, verifying the result in each case, the following : 1. 59 lbs., 7 oz., 14dwts., 19 grs., to grains; and 87400157 grs. to lbs. 2. 56332005 scrs. to lbs. Troy; and 536 lbs. to drams and scruples. 3. 7 tons, 15 cwt., 2 qrs., 16 lbs. to ounces ; and 7563241 drs. to tons. 4. 5838297 oz. to tons; and 38 tons, 17 cwt., 3 qrs., 27 lbs., 15 drs. to dramsr REDUCTION. 103 6. ITlbs., 2 5> 2 3 to grains ; and 84678 grs. Apoth. to oz. Troy. 0. 875 cwt., 2qrs., 15 Ibs.'to stones; and 578421 stones to tons. 7. 3 m., 7 fur., 8 po. to yards ; and 573 miles to inches. 8» 1864428 in. to leagues ; and 74 m., 3 fur., 4 yds. to inches. 9. 4 lea., 2 m,, 2 in. to barleycorns ; and 50 m., 3 po. to yards. 10. 7 fur., 200 yds. to chains ; and 6 cubits, 1 span to feet. 11. 84 yds., 1 qr. to nails; and 56 Eng. ells, 1 qr. to nails. 12. 83 Fr. ells, 8 qrs. to nails ; and 73 Fl. ells, 1 qr. to nails. 18. So ac, 2 ro. to poles ; and 56 ac, 2 ro. to yards. 14. 3 ro., 87 po., 26 yds. to inches ; and 8 ac, 80 po. to feet. 15. 15 ac, 3 ro. to links ; and 50000 po. to acres. 16. 29 cub. yds. to feet; and 158270 cub. in. to yards. 17. 17 cub. yds., 1001 cub. in. to inches ; and 26 cub. yds,, 19 cub, ft. to inches. 18. 563 gals, to pints; and 865843 gills to gallons. 10. 5 pipes, 1 hhd., 35 gals, to pints; and 487634 gills to tierces. 20. 6 hhds., 1 bar. of beer to pints ; and 2307621 pints of wine to hhds. 21. 760 bus., 3 pks. to quarts; and 2 qrs., 1 coomb, 3 pks. to gallons. 22. 3659712 pints to loads; and 7 Ids,, 1 qr., 2 bus. to pecks. 23. 250 chaldrons to bushels ; and 186043 pks. to chaldrons. 24. 56 reams, 19 quires to sheets ; and 52073 sheets of paper to reams. 25. 86 wks., 5 d., 17 hrs, to seconds ; and 1 mo. of 80 days, 28 hrs., 59 sec. to seconds. (7) How many barrels, gallons, quarts, and pints are there in 1336381 half-pints ? (8) One year being equivalent to 365 days, 6 hours, find how many seconds there are in 27 years, 245 days. (9) From 9 o'clock p.bi., Aug. 5, 1852, to 6 o'clock a.m., March 3, 1858, how many hours are there, and how many seconds ? (10) In England there are 50585 square miles ; in Wales, 8125 square miles ; in Scotland, 29167 square miles : how many square acres do they all contain ? COMPOUN-D ADDITIOK 117. Compound Addition is the method of collecting several numbers of the same kind, but containing different denominations of that kind, into one sum. 104 - AlUTHMETIC. Rule. " Arrange the numbers, so that those of the same denomina- tion may be under each other in the same column, and draw a line below them. Add the numbers of the lowest denomination together, and find by reduction how many units of the next higher denomination are con- tained in this sum. Set down the remainder, if any, under the column just added, and carry the quotient to the next column : proceed thus with all the columns." Ex. 1. Add together £2. 4^. 7M., £S. 5s. IQif/., £15. 15s., and £83. 12*. lUd. Proceeding by the Rule given above, £ s. d. 2 . 4 . 7-^ 8 . 5 . loi 15 . 15 . 83 . 12 . m £54 . 18 . 5} Reason for the above process. The sum of 2 farthings. 1 farthing, and 2 farthings, =5 farthings, =1 penny, and 1 farthing ; we therefore put down J, that is, one farthing, and carry 1 penny to the column of pence. Then (l + ll + 10 + 7K = 29(/.-(12x2 + 5y. or 2 shillings, and 5 pence ; we therefore put down 5d., and carry on the 2 to the column of shillings. Then (2 + 12 + 15 + 5 + 4).?. = 38^. = (20 x 1 + 18).?. = £1., and 18.?.; we therefore put down 18^., and carry on the 1 pound to the column of pounds. Then (1 + 33 + 15 + 3 + 2) pounds =: £54. Therefore the result is £54. 18.?. 5-}d. JVote. The method of proof is the same as that in Simple Addition. Ex. 2. Add together 34 tons, 15 cwt., 1 qr., 14 lbs. ; 42 tons, 3 cwt., 181bs. ; 18 tons, 19 cwt., 3 qrs. ; 7 c\vt=, 6 lbs ; 2 qrs., 19 lbs. ; and 3 tons, 7 lbs. tons cwt. qrs. lbs. 84 . 15 . 1 . 14 42 . 3 . . 18 18 . 19 . 3 . . 7.0. 6 . . 2 . 19 8 . 0.0. 7 Ans. 09 . G . . 8 COMPOUND ADDITION. 105 Ex. XXXIV„ £. s. d. £. s. d. £. s. d. (1) 1 . 7 . 6 (2) 25 . 17 . (3) 83 . 10 . 8f G . „ 3 G3 . 15 . 10 G7 . . 7^ 5 . 11 . 4 24 . 19 . 8 73 . 19 . lOj 8.8.8 81 c 17 . 11 29 . 9 . 9J 2 „ 1 . 11 67 c . 3 47 . 16 . 81- £. s. d. £. s. d. £. s. d. (4) 5 . 17 . 10^ (5) C3 . 15 . 2i (6) 528 . 14 . 11| 26 , . ll' 83 . 8 n 854 . 19 . 4 7 , 3 . 41 41 . . "5- 578 . 18 9} 73 . 19 . 8i G » 7 . lOi 507 . OJ 30 . 14 . Jl 76 . 17 . ll 859 . 14 „ 111^ (on;i . cwt. qrs. ibs. oz. drs. EC. grs. ac. ro. po. (7) 16 . 17 2 25 (8) 22 . 3 2 19 (9) 82 . 2 . 24 13 . 10 . . 20 5Q . . 1 . 10 18 . 3 . 14 17 . 15 o 2 . 19 3 . 2 11 20 . 1 . 27 84 . . 3 . 27 15 . 6 , 1 . 9 56 . . 11 . 11 . 1 . 11 79 . 4 . 1 . 10 45 . 3 . 30 (10) Find the sum of £28. 14.?. 6|c?., £27. 18s. ^d., £79. 12.s. Gd., £19. 18s. lOU., and £85. 14^. SJc?. ; also of £678. 10*. 2d., £325. 6s. 5d., £487. 18*. 9d", £507. 0.?. lid, and £779. 10*. 8d. ; also of £568. 10.?. SU., £259. 19*. 5U., £188. 11*. 4:U., £157. 9*. 32^., £13. 13*. d^d., and £779. 8*. 8|rf. ; also of £941. 14*. 2d. ^ £888. 17*. Q^d., £309. 19*. lO^rf., £679. 2*. llfd, £455. 16*., and £447. Os. 7^ ; also of £3966. 16*. 93J., £2. 11*. 7fc?., £3795. 0*. 2ld., £37. 17*. 0|d, £48. 0*. 0^., and £59000. 14*. G|rf. ; also of £6491, £3651 c 10*. 3ld., £8000. 0*. llfd, £5510. 19*. lOU., £50430. 12*. l}d., £316. 14*. 5^d., and £4850. 18*. M. ; also of £306217. 13^. 9^d., £5.5. 0*. 9d., £450812. 15*. 2ld., £9837. 1*. 5U., and £2939. 3*. life?. ; and prove the result in each case. (11) Add together 2 lbs., 9 oz., 1 dwt., 23grs.; 8 lbs., 6oz., 4 dwts., 20 grs.; 1 lb., 10 oz., 5 dwts., 12 grs. ; 14 lbs., 11 oz., 14 dwts., 19 grs.; and 21 lbs., 8 oz., 13 dwts., 11 grs.: also 22 lbs., 7 dwts., 15 grs. ; 15 lbs., 11 oz., 18 grs. ; 34 lbs., 9 oz., 12 dwts. ; 74 lbs., 1 oz., 1 dwt., 20 grs. ; and 46 lbs., 11 oz., 16 dwts., 19 grs.: also 1740 oz., 9 dwts., 19 grs., - 4179 oz., 11 dwts., 14 grs.; 8497 oz., 12 dwts., 22 grs.; 5629 oz., 19 dwts., 17 grs.; and 1038 oz., 4 dwts,, 14 grs. : verify each result. (12) Add together 3 drs., 2 scr., 19 grs. ; 2 drs,, 2 scr., 11 grs.; 7 drsc 106 ARITHMETIC. 17 grs.; 6 drs., 1 scr., 9 grs.; and 5 drs., 1 scr., 13 grs,: also 10 lbs., 8 oz., 4 drs,, 1 scr. ; 66 lbs., 10 oz., 2 drs.; 19 lbs., 9 oz., 8 drs., 2 scr. ; 55 lbs., 6 drs.; and 79 lbs., 11 oz., 4 drs., 1 scr.: also 13 lbs., 6 oz., 7 drs., 2 scr., 17 grs. ; 19 lbs., 11 oz., 1 scr., 18 grs.; 36 lbs., 3 oz., 2 scr., 19 grs. ; 6 oz., 7 drs., 7 grs. ; and 176 lbs., 96 grs. : explain the process in each case. (13) Find the aggregate of 18 lbs., 14 oz., 6 drs. ; 9 lbs., 6 oz., 15 drs.; 45 lbs., 9 oz., 8 drs. ; 9 lbs., 15 oz., 4 drs. ; and 14 lbs., 12 oz., 12 drs. : also of Icwt., 2qrs., 26 lbs., 10 oz.; 11 cwi, 18 lbs., 9 oz.; 13 cwi, 3 qrs., 17 lbs., 14oz.; 7 cwi, 1 qr., 25 lbs., 9oz.; and 19 cwi, 2 qrs., 19 lbs., 14 oz.: also of 306 tons, 15 cwt., 2 qrs., 15 lbs. ; 731 tons, 6 cwi, 3 qrs,, 24 lbs.; 279 tons, 7 cwi, 10 lbs.; 896 tons, 9 cwi, 1 qr., I71bs. ; and 10 cwt., 2 qrs., 16 lbs.: also of 23 tons, 12 cwt., 15 lbs., 12 oz. ; 58 ton?, 17 cwi, 1 qr., 10 oz. ; 67 tons, 8 qrs., 15 oz. ; 19 cwt., 27 lbs. ; and 3 tons, 13 lbs., 13 oz. : prove the results. (14) Find the sum of 11 yds., 2 fi, 9 in.; 27 yds., 1 ft., 3 in. ; S6 yds., 2fi, 10 in.; 48yds., 2ft., 11 in.; and 51 yds., Ifi, 8in.: also of 26 m., 7 fur., 23po., 3 yds.; 22 m., 5 fur., 27 po., 5 yds.; 37 ni., 4 fur., 3 yds.; 86 m., 6 fur., 88 po., 8 yds.; and 25 m., 1 fur., 29 po., 2^yds. : also of 14m., 7 fur., 23 po., 2iyds., 2 fi, 11 in.; 12m., 5 fur., 1 yd., 2 ft., 8 in. ; 27m., 2 fur., 18 po., 8|-yds., Ifi, 10 in.; 36 m., 6 fur., 83 po., 4^ yds., 2fi, 6 in. ; and 75 m,, 1 fur., 21 po., 8 yds., 1 fi, 7 in.: also of 2 lea., 1 m., 3 fur., 103 yds. ; 67 lea., 8 fur., 157 yds. ; 11 lea., 1 m., 98 yds. ; 9 lea., 2 m., 5 fur., 87 yds.; and 34 lea., 2 m., 7 fur., 198yds. (15) Find the sum of 43 yds., 2 qrs., 3na. ; 87 yds., 2 qrs., 1 na. ; 28 yds., 8 qrs., 2 na. ; 41 yds., 2 qrs., 2 na. ; and 88 yds., 2 qrs., 3 na. : and of 11 Eng. ells, 2 qrs., 8 na.; 13Eng. ells, 2 qrs., 1 na.; 89 Eng. ells, 4 qrs., 2 na. ; 37 Eng. ells, 4 qrs., 8na.; and 79 Eng. ells, 8na.: and prove each result. (16) Find the sum of 25 ac, 2 ro., 36 po.; 80 ac, 2 ro., 25 po. ; 26 ac, 2ro., 85 po.; 63 ac, 1 ro., 31 po.; and 84 ac, 2ro., 29 po.: also of 5ac., 2 ro., 15 po., 25| sq. yds., 101 sq. in. ; 9 ac, 1 ro., 85 po., 121 sq. y^g,^ 87sq.in.; 42 ac, 8ro., 24 po., 23i sq. yds., 57sq. in. ; 12 ac, 2ro., 6 po., 13isq. yds., 23sq. in.; and 17 ac, 24 po., 80sq. yds., 113sq.in. : explain each process. (17) Find the sum of 3 c yds., 23 c ft., 171 c. in. ; 17 c yds., 17 c. ft,, 81 c in. ; 28 c yds., 26 c fi, 1000 c. in. ; and 84 c yds., 28 c fi, 1101 c in. : also of 12 po., 18 sq. yds., 7 sq. ft., 85 sq. in.; 13 po., 243- sq. yds., 8 sq. ft., 63 sq. in.; 14 po., 29^ sq. yds., 5 sq. ft., 131 sq. in.; 15 po., 19 sq. yds., 3 sq. ft., 126 sq. in.; and 16 po., 28} sq. yds., 130 sq. in. (18) Add together 39 gak, 8 qts., 1 pi; 48 gals-, 2 qts., 1 pii COMPOUND ADDITION. 107 66 gals., 1 pt. ; 74 gals., 3 qts. ; and 84 gals,, 3 qts., 1 pt. ^ also 2 pipes, 42 gals., 3 qts.; 3G gals., 1 qt. ; 5 pipes, 48 gals. ; 12 pipes, 53 gals., 3 qts.; and 27 pipes, 2 qts., of wine : also 19 hhds., 10 gals., 3 pts. ; 29 hhds., 50 gals., 7 pts. ; 116 hhds., 46 gals., 5 pts. ; 2 hhds., 2 pts.; and 235 hhds., 1 bar., 3 qts., of beer. (19) Add together 14 qrs., 6 bus., 3 pks., 7 pts. ; 37 qrs., 5 bus., l3pts. ; 43 qrs., 2 pks., 14 pts. ; 57 qrs., 7 bus., 8 pks., 12 pts. ; and 106 qrs., 4 bus., 13 pts. : also 87 Ids., 37 bus., 2pks. ; 92 Ids., 24 bus., Spks.; 136 Ids., 28 bus., Ipk.; 157 Ids., 36 bus., 2 pks. ; 540 Ids., 1 pk.; and 786 Ids., 89 bus. (20) Add together 4 mo., 3 w., 5 d., 23 h., 46 m.; 5 mo., 1 d., 17 h., 57 m. ; 6 mo., 2 av., 1 h.; 1 w., 6 d., 23 h., 59 m. ; and 11 mo., 1 w., 58 m. : also 7 yrs., 28 w., 3 s.; 26 yrs., 5 w., 5 d. ; 58 yrs., 6 d., 23 h., 59 s. ; 43 w., 28 h., 50 m., 12 s.; and 124 yrs., 14 w., 19 h., 37 s. (21) When B was born, A's age was 2 yrs., 9 mo., 3 w., 4d. ; when C was born, B's age was 18 yrs., and 3 d. ; when D was born, Cs age was 9 mo., 2 w., 3d., 23 h. ; when E was born, D's age was 6 yrs., llrao., 23 hrs. ; when F was born, jE"s age was 7 yrs., 3 w., 5 d., 15 h. What was A\ age on i^'s 5th birth-day ? 118. If other fractions of a penny, as well as those which denote farthings, be involved, the process is exactly the same as the above ; those fractions being first added together by the ordinary rule of Addition of Fractions. For example, add together, £11. 4*. 6\d. ', £12. Zs. Tjjjd.; £4. 7s. 8f rf. ; £o. 3s. 2|c?. ; and £G. Ws. OU. £. s. d. 11 . 4 . 51 Now(J- + J^ + |- + | + *)rf. 12 . 2 . 7^^ =(i+^+f+jy-=a+i-+#^+MK 4 . 7 . 8f =(i+i^d.^md. 5 . 3 . 2|- we therefore put down xf^-> carry on 1 to the 6 . 10 . OA- column of pence, and proceed by Rule, Art. (117). £39 . 7 . m Ex. XXXV. (1) Add together £2. Os. Ttd. ; £12. 16^. O^d. ; £4. 145. 8---J. ; £10. 0^. aid. ; £1. 75. 5|c?, ; and £14. 155. l^d. (2) Find the sum of £20. 165. b^d. ; £14. 155. Ofd ; £5. 135. ^d. ; £38. 195. lid. ; and £18. 35. 4fd (3) Find the sum of £1. 8*. 6§d ; £2. 45. 7^. ; £3. 6s. ^d. ; £4. 95. V^d.; and £G. 165. 6*1^. 108 ARITHMETia (4) Add together £23. 6s. O^d. ; £4. 0^. 9U. ■ £57. 17s. SJfd } £dG. 19s. Uj^^d. I and £157. 7s. 7lld. (5) Add together £273. 16^. 7lld. ; £370. 11*. 8id. ; £621. 13*. 9J(/o ; £197.45. life?.; and 5^d. COMPOUND SUBTBACTIONo 119. Compound Subtraction is the method of finding the difference between two numbers of the same kind, but containing different denomu nations of that kind. Rule. " PLace the less number below the greater, so that the num- bers of the same denomination may be under each other in the same column, and draw a line below them. Begin at the right hand, and subtract if possible each number of the lower line from that which stands above it, and set the remainder underneath. But when any number in the lower line is greater than the number above it, add to the upper one as many units of the same denomination as make one unit of the next higher denomination; subtract as before, and carry one to the number of the next higher denomination in the lower line ; proceed thus throughout the columns." Ex. 1. Subtract £88. I8s. S^d. from £146. 19*. 6ld. Proceeding by the Rule given above, £. s. d. 146 . 19 . 61 88 . 18 . 8^ £58 . . 9| Reason for the above process. Since ^d. is greater than -}c?., we add to ^d. 4 farthings or 1 penny, thus raising it to 5 farthings ; and when 2 farthings are subtracted from 5 farthings, we have 3 farthings left ; we therefore place down ^d. : and in order to increase the lower number equally with the upper number, wo add 1 penny to the 8 pence. Now 9 pence cannot be taken from 6 pence; we therefore add 12 pence or 1*. to 6 pence, thus raising the latter to 18c?. : we take the 9d. from 18^., and put down the remainder 9c/. ; then adding 1*. to 18*., the latter becomes 19*.: 19*. taken from 19*. leave no remainder; we then subtract £88. from £146., as though they were abstract numbers. It is manifest that in this process, whenever we add to the upper line, we also COMPOUND SUBTRACTION. lOD add a number of the same value to the lower line, so that the final differ' ence is not altered. Ex. 2. Subtract 106 lbs., 11 oz., 16 dwts., from 1-U lbs.. 8 oz., Udwts. lb. 02. d^-is. 1J4 . 8 . 14 106 . 11 . in 8 . 13 Ex. XXXYl. £. f. d. £. f. d. (1) 4.3 . 11 . 5 (2) 149 . 4 . 6J 23 . 2 . 7 86 . 13 . 2^ (3) 309 13 Hi- 119 19 10^ (5) 343 18 5i 11 18 .51 (^) 5875 . 4986 . 19 0^ (6) 663 5 111 349 19 n cwL qr. lbs. oz. fur. po. yds. (7) 63 . . 18 . 1 (8) 14 . 34 .' 5 58 . 1 . 12 . 10 1 . 38 . 4 ac. ro po. qrs. bus. pk. gaL (9) 63 . 1 . 29 (10) 64 . 3 . 1 . 57 . 2 . 38 8.5.3.1 (11) Subtract £456. los. ll^d. from 1-534. 13^\ lOW. ; and prove the result. (12) Find the difference between the following numbers, and verify the results : 1. 426 lbs., 8 02,, 1 dwt., 7 grs., and 888 lbs., 3 oz., 11 dwts., 21 grs. 2. 5836 lbs., and 4976 lbs., 7 oz., 1 3 dwts., 19 gts. 3. 26 tons, 2 qrs., 23 lbs., and 19 tons, 3 cwt., 3 qrs., 18 lbs. 4. 806 tons, 14 cwt., 7 lbs., and 789 tons, 16 lbs. 5. 144 lbs., 9 oz., 4 drs., 1 scr., and 120 lbs., 7 drs., 3 5cr. 110 ARITHMETIC. 6. 418 yds., 1 qr., 1 na., and 887 yds., 8 qrs., 8 na. 7. 15 yds., 1 ft., 5 in., and 18 yds., 2 ft., 7 in. 8. 99 yds., and 87 yds., 1 ft., 11 in. 9. 18 m., 6 fur., 35 po., 8J yds., and 12 m., 88 po., 4 yds. 10. 85 lea., 4 fur., 23 po., 4 yds., 1 ft., and 28 lea., 5 fur., 89 po., U yds., 2 ft. 11. 56 ac, 2 ro., 34 po., and 48 ac, 8 ro., 88 po. 12. 8 ro., 28 po., 27 sq. yds., 7 sq. ft., and 1 ro., 89 po., 28^ sq.yds., 8 sq. ft. 13. 87 cub. yds., 18 cub. ft., 857 cub. in., and 35 cub. yds., 24 cub. ft., 1280 cub. in. 14. 203 tuns, 19 gals., 3 qts., 1 pt., of wine, and 187 tuns, 1 hhd., 29 gals., 2 qts. 15. 88 bar., 2 fir., 7 gals., of beer, and 77 bar., 2 fir., 8 gals., 29 qts. 16. 28 Ids., 2 qrs., 5 bus., 8 pks., and 18 Ids., 2 qrs., 6 bus. 17. 216 yrs., 9 mo., 2 w., 4 d., and 217 yrs. 18 The latitude of St Peter's at Rome is 41°, 58', 54" north, and that of St Paul's at London is 51°, 80', 49" north. Find the difference of their latitude. 19. What sum added to £947. 195. 7|(^. will make £1000 ? 20. A furnished house is worth £4759. 105. 9^-cf. : unfurnished, it is worth £1494. lis. 9^d. By how much does the value of the furniture exceed the value of the house ? 120. If other fractions of a penny than those which denote farthings be involved, ^\e must apply Rule, Art. (76), in order to find the difference of the fractions, and then proceed by Rule, Art. (119). Ex. 1. Subtract £9. 145. 6|c/. from £14. O5. d^d. £. s. d, 14 . . 5J 9 . 14 . 6^ {k-l)d'-^d.^¥, £4.5. llj Ex. 2. Subtract £7. 155. 7{id. from £10. O5. 0|f/. \h is greater than |, therefore we add 1 to |, w^hich makes it y . We must repay the Id. by adding Idi to 7d. £. s. d. 10 . . o| 7 15 . 7-R £2. 4 . ^ r COMPOUND MULTIPLICATION. Ill Ex. XXXVII. Find the difference between (1) £3. 13s, 9id., and £2. 15s. d^d. (2) £20., and -£15. 165. 0?^. (8) £23. 13s. 1\d., and £19. 19^. 7f^. (4) £-116. 10.'. hid., and £305. 11*. 9fc?. (5) £21G3. Is. 7{^d., and £864. 2^. 5\^d. COMPOUND MULTIPLICATION. 121. Compound Multiplication is the method of finding- the amount of any proposed compound number, that is, of any number composed of different denominations, but all of the same kind, when it is repeated a given number of times. Rule. *' Place the multiplier under the lowest denomination of the multiplicand ; multiply the number of the lowest denomination by the multiplier, and find the number of units of the next denomination con- tained in this first product ; if there be a remainder, place it down, adding on the number of units just found to the second product ; for this second product, multiply the number of the next denomination in the multipli- cand by the multiplier, and after carrying on to it the above-mentioned number of units, proceed with the result as with the first product; cany this operation through with all the different denominations of the multi- plicand." Ex. Multiply £5G. 4^. GU. by 5. Proceeding by the Rule given above, £, s. d. 66 . 4 . 61 5 £281 . 2 . 81 Reason for the above process. \d. multiplied by 5 is the same as (| + ^ + -J- + i^ + V)d. = 5 half-pence = 2M.'y we therefore put down W., and carry on 2d. to the denomination of pence : 6d multiplied by 5 = 30(7. ; therefore (2 + G x b)d. = 32d. = (2x12 + 8)(/. -2s.-\-M. ; we therefore put down 8c^., and carry on 2*. to the denomina- tion of shillings : 112 ARITHIvIETIOi 45. multiplied by 5 = 20^. ; therefore (2 + 4x5>. = 22.s. = (20 + 2>. = £1 + 2*. ; we therefore put down 2s., and carry on £1 to the denominc- tion of pounds s Now by Simple Multiplication i;S6x5-£280; therefore £(l4-6Gx5) 2= £(1 + 280) = £281. Therefore the total amount is £281. 2s. 8|^. 122. When the multiplier exceeds 12 it will be the easiest method to split the multiplier into factors, or into factors and parts : thus 15=S x 5; 17 = 3x5 + 2; 23 = 4x5 + 3; 240 = 4x6x10: and so on. Ex. Multiply £55. 12s. 9ld. by 23. £. s. d. 55 . 12 . 9|. 4 222 . 11 . 1 = value of £55. 12s. 9|-d multiplied by 4. 5 1112 .15.5= value of £222. 11*. Id. multiplied by 5, or of £55. 12s. 9}d. multiplied by (4 x 5, or 20). 166 . 18 . 3| = value of £55. 12*. Q^d. multiplied by 3. £1279 . 13 . 8| = value of £55. 125. 9ld. multiplied by (20 + 8), or 23. 2Vote 1. When the multiplicand contains farthings, if one of the factors of the multiplier be even, it will often be advantageous to use it first, as the farthings may disappear. JVote 2. Should the multiplier consist of many factors, it wiU be found in that case convenient to reduce the multiplicand to the lowest denomi- nation contained in it, then to multiply this result by the multiplier, and then to reduce the result back again. Ex. XXXVIII. Multiply (1) £11. ISs. Gd. separately by 2 and 5. (2) £2. 18s. 7U. separately by 4 and 6. ' . . (3) £1. 16.?. 6i(/. separately by 7 and 9. (4) £2. 15s. 2'id. separately by 5 and 8, (5) £3, 16s. O-ld. separately by 11 and 12. (6) £7. 195, 7U. separately by 10 and 12- (7) £347. 15s^9}d. separately by 3 and 11. (8) £583. 05. lOd. separately by 13 and 16. COMPOUND MULTIPLICATION. 113 (9) £1875. 13*. n^d. separately by 21 and 64. (10) £721. 0*. old. separately by 81 and 9G. (11) £5072. 125. SU. separately by 112 and 128. ' (12) £1100. Us. Qld. separately by 62, 82, and 93. (13) £2579. 0*. Old. separately by 147, 155, 474, and 2331. (14) 86 lbs., 7 oz., 16 dwts., 11 grs. separately by 8 and 36. (15) 3 tons, 27 lbs., 13 oz. separately by 11 and 76. (16) 45 lbs., 7 oz., 3 drs., 2sc. separately by 12 and 68. (17) 67 yds., 1 qr., 2 na. separately by 9 and 53. (18) 70 yds., 2 ft., 10 in. separately by 7 and 29. (19) 67 re, 38 po., 27 yds., 2 ft. separately by 11 and 112. (20) 380 ac, 3 ro., 32 po. separately by 12 and 106. (21) 57 gals., 3 qts. separately by 10 and 257. (22) 76 qrs., 5 bus., 2 pks. separately by 13 and 240. (23) 5 wks., 6 d., 18 h., 14 m. separately by 11 and 339. (24) 84 hhds., 43 gals., 1 pt. of wine separately by 27 and 364. (25) 43 bar., 13 gals., 1 qt., 1 pt. of beer separately by 39 and 764. (26) A person buys 67 lambs at £1. 0*. d^d. each ; 73 sheep at £2. 2s. Hid. each ; 12 cows at the average of £37. Os. ^^d. for every 3 of them ; and 17 horses at 37 guineas each : the expenses of getting them all home amount to 17A- guineas. What money must he draw from his bankers to pay for the whole outlay ? (27) There are 7 chests of drawers : in each chest there are 18 drawers ; and in each drawer 8 divisions ; and in each division there is placed £16. 6*. 8c?. How much money is deposited in the chests ? 123. If the multiplicand contain, instead of farthings, some other fraction of a penny, the process is exactly the same as the above : thus, Ex. 1, if wc liad to multiply £22. 15.?. 4^(^. by 43 .; £. s. 22 . 15 d. . 4| 8 43 = 5x8 + 3 182 . 8 e 1 5 foY^^d.xS = i£d, = 5d. 010 . 15 68 . 6 forff/. x3=\V. = 1|g?. £979 c 1 *^ 114 ARITHMETIC. Ex. 2. Multiply £36. 10*. 0]^d. by 231. 231 = 7x33 = 7x3x11. £. s. d. 8G . 10 . 01^ 7 265 . 10 . 4jV for^dx7=f^<;. = 4xV- 8 766 . 11 . OjV iovfjd.-K^^^jd. 11 £8432 . 1 . 3if for {\d. x 11 - ^d. -= 3}| ^^^^+ 100— _ 1250.. "^^^^"lor ==£53 + 12-50.. COMPOUND DIVISION. 119 = £5S + 12s. + 6d. = £53. Us. ed, Ex. 2. Divide £1668. 15*. by 1500. 1500 = 3x5x100; first divide by the factors 3 and 5, and then by 100 : it will bo found best in all cases of this kind to do so. £. s. 3 1 1668 . 15 556 £1-11 . 5 20 2-255. 12 800d, Therefore the quotient is £1. 2^. 3d. Ex. XLII. £396. 95. 2(/.-^10. (2) £2025-:- 1000. (4) £262.105.-2400. (6) 21 ac, 3 ro., 17 perches x "02 ; and £375. 35. 24 ac, 3 ro., 10 perches x 112, and x 11-2. When the divisor and dividend are both compound numbers of the £1787. 105. -f- 100. £1447. 185. 4d-f- 1000. £26380. 45. 2rf.-- 25000. x -0507. (1) (8) (5) (7) (8) 127. same kind. Rule. " Reduce both numbers to the same denomination : divide as in Simple Division, and the result will be the answer required." Ex. How often is 5s. 3f c?. contained in £15. Proceeding by the above Rule, s. d. £. 5 . 31 15 . 12 20 63 318 4 12 255 8825 4 15300 255) 15800 (60 1530 • Therefore 60 ig the answer. 9rf. ? 18 . 9 120 AHITHMETIC. Reason for the ahove process. 5s. S^d. = 255 farthings, £15. 185. 9a?. = 15300 farthings; and 255 farthings subtracted 60 times from 15800 farthings leave no re- mainder. Ex. XLIII. (1) £2. 12s. Sd,~ls. 4^d, (2) £55. 185. lOid -- £2. 8s. 7|c?. (3) £160. 45. 81c?. -^ £1.105. ejd (4) £401. 45. 3d. -:-£2. II5. SJg?. (5) 44 cwt, 2 qrs., 11 Ihs. -- 1 cwt., 2 qrs., 17 lbs. (6) 272 yds., 1 qr.-^7 yds., 2 qrs., 1 na. (7 J 9487 bus., 2 pks. -f- 143 bus., 8 pks. (8) 1416 ac., 2 re, 16 po. -^ 4 ac, 3 re, 27 po. (9) 57 lea., 1 mi., 956 yds. -^ 7 fur., 87 yds., 1 ft., 5 in. (10) 617 Ids., 1 qr. -- 12 qrs., 1 pk. 128. We shall now add some examples of the Multiplication and Division of numbers, comprising different denominations, but of the same kind, by mixed numbers. In the case of IMultiplying by a mixed number, it will generally be found advantageous, first to multiply by the integral part, and then to add to the result thus obtained the result given by multiplying by the fractional part. Thus, for example : ^lultiply £2. 65. 8t/. by S^. (£2. 65. 8fi?.)x8-£7. (£2.05. 8ri.) x7 „ £16. 65. 8d 10 10— = ^^- ^^*- ^'^• Thel-efore (£2. Gs. Sd.) x S^^-=£7 + £l. I2s. 86?. = £8. 125. 8^?. In Division it will be found advantageous to reduce the mixed number to an improper fraction. Thus, for example : Divide .-€89. 1/5. 62d by 19fd Now £89.17.6iA^V = ^ ^''-y'^)''^ ,. MISCELLANEOUS EXAMPLES WOllKED OUT. 121 £. s. d. 89 , 17 . G£ 4 79) 859 . 10 . 3 {4£, S16 43 20 79) 870 (116-. 79 80 79 1 12 79) 15 (Od. Therefore the quotient is £4. 11^. 0]^d. Ex. XLIV. (1) £18. 12*. Il|-. = (8xl2)c?.; therefore -— — -—-, or — — , or 93J = the number of gallons which the 8x 12 4 ^ cask must contain, in order that its contents may be sold at 85-. a gallon. MISCELLANEOUS EXAMPLES WOEKED OUT. 123 Therefore (93f — 60), or S8j-tlie number of gallons of water which have to be added. Ex. 7. How many yards of cloth, worth Ss. 7-hd. a yard, must be given in exchange for 144 yards of cloth, worth 18^. lid. a yard? The value of 144 yards at 18^. l^d. a yard, = {18s.lld.)xlU, = £130,10^. = 62640 half-pence; and 3*. ^J^d. - 87 half-pence ; therefore the number of yards required = " _ = 720. Oi- thus, since 18*. lic?. = (3*. 7Jc?.) x 5, it is clear that the number of yards required = 144 x 5, = 720. Ex. 8. A traveller walks 22 miles a day, and after he has gone 84 miles another follows him at the rate of 34 miles a day ; in what time will the second traveller overtake the first 1 The second traveller has to walk over 84 miles more than the first before he can overtake him. Each day he W'alks (34—22) or 12 miles more than the first ; therefore f|- or 7 is the number of days required. Ex. 9. A mixture is made of 8 gallons of spirits at 12*. lOd. a gallon, 7 gallons at 10*. 6d. a gallon, and 10 gallons at 9*. Id. a gallon ; at wliat price per gallon must the mixture be sold, 1st, that the seller may neither gain nor lose by his bargain; 2nd, that he may gain £1. 13*. by it ; 3rd, that he may lose 7 guineas; and 4th, that he may reserve 10 gallons of the mixture for himself, and sell the remainder so as to realize the money he laid out ? £. s. d. 8 gallons at 12*. Wd. cost 5.2.8 7 gallons at 10*. 6d. cost 3 . 13 . 6 10 gallons at 9*. Id. cost 4 . 10 . 10 therefore 25 gallons cost £13 .7.0 1st. If he is neither to gain or lose, he must sell 1 gallon for ^ OR ~' ; wliich, worked out, gives 10.?. 80^3^-. as the price required. 124 ARITHMETIC. 2nd. If he is to gain £1. 13*. 25 gallons must be sold for £13. 7*. + £1. ISs., or £15 ; £15 tlierefore, 1 gallon must be sold for -^_- ; which, worked out, gives 12*. as the price required. 3rd. If he is to lose 7 guineas, 25 gallons must be sold for £13. 7s.- £7. 7s. or £6; therefore 1 gallon must be sold for ^ ; which, worked out, gives 4:8. 9^d. fq. as the price required. 4th. If he is to retain 10 gallons for his own use, 15 gallons must be sold for £13. 7^.; £18 7s therefore 1 gallon must be sold for — -^ — '-; which, worked out, gives 17*. 9^d. ^q. as the price required. Ex. 10. A club, consisting of 56 persons, joined for a lottery ticket of 12 guineas value, and it came up a prize of £7000 : what sum did each Qian contribute, and what did each man gain ? 66 persons subscribe 12 guineas ; therefore each person subscribes — ~^ , 50 or 4*. Gd. 56 persons receive a prize of £7000 ; ,, ^ , . £7000 therefore each person receives —z^r-, 5d or £125 ; tlierefore each person gains £125. — 46\ Gd. = £124. 155. Gd. Ex. 11. Divide £20 among A B, and C, so that B may have 2 guineas more than A, and that C may have 2^. less than B. Now JB's share — A's share -f £2. 2s, C 's share — ^'s share — 2s. = ^'s share + £2. 2s. - 2s. ■-= A's share + £2. But, by the question, ^'s share + B's share + Cs share = £20, 'jT A'a share + (A's share + £2. 2*.) + (^'s share 4- £2) = £20, DECIMAL COINAGE. 125 or 3 times A's. share + £4. 2s. = £20 ; therefore evidently 8 times ^'s share = £20 - £4. 2s. = £15.185., ^, , £15. 185. -. _^ or ^ s share = • ^ = ^^- ^^• B's share = £7. 8*. C's share = £7. Gs. Ex. 12v Divide £8. ll*. Gd. among 5 men, 6 women, and 7 hoys; Diving each woman twice as much as each boy, and each man thrice as mach as each woman. Since each woman's share = twice each boy's share, therefore 6 women's shares = 12 boys' shares. Again, since each man's share = thrice each woman's share, therefore, 5 men's shares =15 women's shares, = 80 boys' shares, but 5 men's shares + 6 women's shares + 7 boys' shares = £8. 11*. Gd., or 80 boys' shares + 12 boys' shares + 7 boys' shares = £8. 11 5. Gd., or 49 boys' shares = £8. lis. Gd. = 843 sixpences. Therefore, each boy's share = — sixpences, = 7 sixpences = 3s. Gd. Therefore, each ^voman's share = 7-?., each man's share = £1. 1*. Od. DECIMAL COINAGE. 129. It may be well to notice here some of the advantages which would result from a decimal comage of pounds, florins, cents, and mils ; the pound being of the same value as the pound sterling at present ; the florin being =Joth of £1 ; the cent being =Joth of a florin, or^yi^th of £1 ; the mil (m.) being = Joth of a cent, or = j^-th of a florin, or= j-J^oth of £1 . The Table would stand thus : 10 Mils make 1 cent, 1 c. 10 Cents 1 florin, 1 fl. 10 Florins 1 pound, £1. 180. In such a system, much of the labour of reducing superior to inferior denominations, and the converse, would be done away with ; for M^e could at once say, £24. 3 fl. 7 c 2 m. = 24372 m. Since by performing the operation of reduction at length, we obtain 126 AKITIIMETIO. £. fl. c. m. 24 . 3 . 7 . 2 10 240 + 8; or 243 fl. 10 2430 + 7, or 2437 c. 10 24370 + 2, or 24372 m. or we might say £24. 3 fl. 7 c. 2 m. = £24-372 ; ^24000 + 300 + 70 + 2 1000 24372 "1000 = £24-372. SimHarly, £24. 3 fl. 7 c. Conversely 24372 •, proceeding by Rule (Art. 116 10 2 m. = 243-72 fl., or = 2457-2 a mils = £24. 3fl. 7 c. 2 m., ), we get 24372 10 2437-2 m. 10 243 -7 c. 24-3fl. hence 24372 m. = £24. 3fl. 7 c. 2 m. ; or we might say 24372 m. = £24-372 ; for 24372 m. = £^^ - £24-372. Similarly 24372 m. = 245-72 fl., or = 2437-2 Co Again, £18. 3 fl. 9 m. = 18309 m., or, proceeding by Rule (Art. 116), £. fl. m. 18 . 3 . 9 10 TSO + S, or 183fl. 10 1830 c. 10 183004 9, or 18309 m. DECIMAL COINAGE. 127 or we might say £18. Sfl. 9 m. = £18-309 ; for£18. 3fl.9m.==£(l84-^4-^ + 3^) '180G0 + 8C0 + 9\ £ V 1000 18309 10 10 ! i 18809 18S0- 10 j 183- " 1000 ==£18-809. Similarly £18. Sfl. 9 m. = 183'09fl., or=1880-9c. Conversely 18809 m. = £18. 3 fl. 9 m. for, proceeding by Rnle (Art. 116), we get ■9m. Oc. 18-3fl. or 18809 m. = £18. 3 fl. c. 9 m. Similarly 18309 m. = £^^^-£18-309. or 18309 m. = 183-09 fl., or = IBSO'O c. Again, £254. 5^0. = £254. 5*5 c. 100 25400 C. + 5-5 c. = 25405-5 c. = 254055 m. Also, £254. 5ifl. = £254. 5-25 fl. 10 2640 fl. + 5-25 fl. = 2545-25 fl. = 25452-5 c. =254525 m. Ex. XLV. Reduce, expressing in each successive inferior denomination and verifying each result : (1) £15. 6fl. to mils, and 6 fl. 8c. 2 m. to mils. (2) £30. 91 fl. to mils, and £96. 1 fl. 2 c. 9 m. to mils. 128 ARITHMETIC. (3) X18. 6^c. to mils, and 9Jfl. to mils. (4) £10. 1 m. to mils, and £46. 2| c. to mils. 131. The addition, subtraction, multiplication, and division of money would also be much simplified by the adoption of a decimal coinage, as will be evident from the following examples. Ex. 1. Find the sum of £18. 6 fl. 3 c. 5 m.; 9fl. 9 m.; £24. Im.; 8c. 2m.; 5^- m. £18. 6 fl. 3 c. 5 m. = 18635, £. or= 18-635, 9fl. 9m.= 909, or= -909, £24. lm. = 24001, or= 24-001, 3 c. 2m.= 32, or= -032, 5Jfl.= 52.5, or= -525, 44102 ra., or = £44-102, each of which results = = £44, Ifl. 2 m. Ex. 2. From £16. 3 c. 2 m., subtract £14. 4 fl. 9 m. m. £16. 3 c. 2 m. = 16032, £14. 4 fl. 9 m. = 14409, £. or= 16-032, or =14-409, 1628 m., or = £1023, each of wdiich results = £1. fl. 2 c. 3 m. Ex. 3. Multiply £16. 3 c. 2 m. by 23. £16. 3 c. 2 m. = 16032 m., or = £16-0S2. m. £. 16032 ■ 16-032 23 23 48096 48096 32064 32064 868736 m. £868-786 each of the above results = £868. 7 fl. 8 c. 6. ra. Ex. 4. Divide £368. 7 fl. 8 c. 6 m. by 23. In other words, divide 868736 m. by 23, or £368-736 by 23. DECIMAL COIJS'AGE. 129 tn. m. £, £,, 23) 8G873G (1G032 23) 3G8-73G (1G-U32, '23 23 138 138 138 138 73 73 46 46 46 46 each of the above results = £16. Ofl. Sc. 2in. Note. Similar advantages would result from tlie use of a decimal system in weights and measures, Ex. XLVI. 1. Add togetlier (1) £7G. 8 f 1. 5 c. 3 m. ; £27. 9 fl. m. ; £84. Ic. ; £56. 3 £L G c. 2m. ; £19. Im. (2) £252. 2.\ fl. ; £300. 2^ c. ; ^\ fl. ; b\ c. 2. Find the difference between (1) £19. 5 fl., and £10. 3 fl. 9 c. (2) £20, and £19. 9 fl. 9 c. 9 m. (3) £5. b\ fl., and £4. 4i- c. 3. Multiply (1) £76. 8 fl. 3 m. separately by 5 and G3. (2) 9 fl. 2i c. separately by 18 and 1008. (3) £150. 5 m. separately by 2005 and 18576. 4. Divide (1) £194, 5 fl. 7 c. 5 m. by 5. (2) £10764. 2 fl. 4 m. by 11. (3) £342130. 8 fl. by 7380. Ex. XLVI I. Miscellaneous Questions and Examples on Arts. (100 — 131). A'ofe.— Where the contrary is not expressed, a year is supposed to consist of 365 days. I. (1) Explain the meaning of the term 'Reduction.' Reduce 537983 half-guineas into seven- shilling pieces, and also into groats. 9 130 ARITHMETIC. (2) "What is the standard of the gold, and silver, and copper coinage in this kingdom ? According to the present law in England for what sums respectively are copper and silver legal tenders ? (3) \[^mt is meant by ^Compound JNIultiplication'? Can concrete numbers of the same or different kinds be multiplied together? Give the reason. What is the cost of school accommodation for 13750 chil- dren at £1. 18s. 6M. each? (4) How many nobles are equivalent to £26. 14-?. ? (5) A person bought 1763 yards of cloth at 5s. S^d. per yard, and retailed it at 6s. lid. per yard : what was his profit? (6) A person's weekly income is £14, and his quarterly expenditure is £128. 10*. ; how much will he have saved at the end of 8 years ? (supposing a year to consist of 52 v/eeks). (7) An equal number of guineas, pounds, half-guineas, crowns, and half-crowns amount to £898. 5s. i how many of each sort are there ? (8) What quantity cf water must I add to a pipe of wine, which cost £90, to reduce its price -to 10.?. a gallon ? II. (1) Explain the meaning of ^Compound Division*: what different cases are there of it ? If £1844. 2,?. 8^d. be divided equally among 40 persons, how much Will each receive ? (2) A house and its furniture are worth £6734. 5^. Qd. ; but the house is worth 8 times as much as the furniture ; what is the house TTorth ? (3) Define 'a squnro ', 'a cubo ' ; shew clearly by a figure how many cubic feet there are in a cubic yard. Reduce 4208239040 cub. in. to cub. yds. ; and find how many grains of wheat there are in a load, if a pint contains 7000 grains. (4) Divide £3. 13^. Od. between two j}ersons, so that one shall receive half as much again as the other, (5) A jeweller sold jewels to the value of 834 guineas, for which he received in part 1429 dollars, worth 48. 6d. each ; what sum remained unpaid ? (6) The tax on a certain property amounts to £974. I65. S}d. at tlio rate of 2.?. 2|f/. in the pound. What is the value of the property? (7) If I bottle off two-thirds of 2 pipcG of wine into quarts, and tha ^rest into pints, how many dozens of each shall I have? (8) A servant's wages are £10. Qs. a year; how much ought he to receive for 7 weeks? (supposing a year to consist of 52 wecks> MISCELLANEOUS QUESTIONS AND EXAMPLES. 131 III. (1) What are the different uses to which Troy weight and Avoir- dupois weight are respectively applied ? Express 66 Ihs, Avoirdupois in lbs. &c. Troy. (2) A factor bought 5G pieces of stuff for £1569. 17^. Ad. at 4s. lOd. a yard : how many yards were there in each piece ? (3) How many farthings are there in 5 half-sovereigns, 5 half-crowns, 5 sixpences, and 5 half-pence ? (4) Goods are bought at 6M. per Ih., and the cost of carriage is lid. per lb, ; they are sold at £4. 10s. per cwt. : what is the gain or loss per cwt. ? (5) What is meant by a ' mean solar day ' ? How does the ^ solar ' year differ from the ' civil ' year ? State clearly the methods which have been adopted to correct the error arising therefrom, (6) A gentleman laid up in the year 1851 £294. Is. 6d., having spent daily £1. 12s. Gd. : what was his income in that year? (7) Divide 198 guineas among 4 persons, so that the second may have twice as much as the first, the third 3 times as much as the second, and the fourth 4 times as much as the third. (8) A person with £5. 7 florins, 9 cents, and 1 mil in his pocket, goes to the sea-side for 2 days : he spends in Railway fare 6fl. 2 c. 5 m.; in cab fare 1 fl. 2 c. 5 m. ; and his Hotel bill is 13 fl. 5 c. "VYliat sum does he return home with ? IV. (1) What are the standards of weight and capacity in England, and how are they fixed ? (2) Two persons buy postage-stamps at 12 a shilling ; one retails them at 11 for a shilling, and the other at 13^. for a dozen ; compare the gains on selling the same number of stamps. (3) How many Rubles at 3*. 4^. each are equal in value to 870 Napoleons, at 15*. 9f^. to the Napoleon ? (4) A hundred sovereigns all equally light; are worth ninetj'-fivo pounds ; what is the value of each in shillings ? (5) Find 1. The sum of £27. 8 c. 9 m. ,' £5G0. 2 J fl. ; £30. 8 c. 7 m. 2. The quotient of £405. 5 fl. 3 c. 6 m. by IG. (6) A person lays out £43. 9*. 4d. in spirits at 5*. 4d. a gallon ; some of which leaked out in the carriage ; however, he sold the remainder for £54, at the rate of 7*. Qd. a gallon : how many gallons leaked out ? (7) If a piece of ground contam 24 acres, and an inclosure of 17 9-2 132 ARITHMETIC. acres, 3 roods be taken out of it, how many perches are there in the remainder ? (8) How many hours have elapsed since the birth of Christ to the year 1852, supposing each year to consist of 365 days, 6 hours ? V. (1) Explain how the statute defines 'a yard', with reference to a natural standard of length. Find the corresponding linear unit, when an acre is one hundred thousand square units. (2) How many barley-corns will reach round the earth, supposing the circumference of it to be 25000 miles ? (3) If a single' article cost 8s. Id., how many dozens can be bought for £86. \Qs. ? (4) A bankrupt owes £3549, and can pay 17*. Qd. in the pound. "NVhat are his effects worth, and what loss do his creditors sustain ? (5) A piece of money is worth 16s. 8d. ; how many guineas are there in 253 such pieces ? (6) How many times will a pendulum vibrate in 24 hours, which vibrates 5 times in 2 seconds 1 (7) If the sum paid for 247 gallons of spirit amount, together with the duty, to £619. lis. 2d. ; and the duty on each gallon be ^th part of its original cost ; what is the duty per gallon ? (8) 12 persons on a journey each spend £23. 4 c. 6 m, in board and lodging ; 6 of them agree to pay the travelling expences, the share of each amounting to £18. 1 m. Find the amount of expenditure during the journey. VI. (1) WTiiat is the meaning of the word 'Carat ' as applied to gold, and as applied to diamonds? How many * carats' fine is standard gold? If from 2793461 lbs. Troy of gold there be corned £130524465. 4*. 6rf., find the value of each lb. (2) A wheel makes 514 revolutions in passing over 1 mile, 467 yards, 1 foot : what is its circumference ? (3) How much must I pay for 455 Napoleons, a Napoleon being worth 165. 4.ld. ? (4) A grocer buys a hogshead of sugar, containing half a ton, for £30, and retails it at "Jld. per lb. ; how niuch money does he make ? (5) A merchant buys 10 gallons of spirit at 12*. a gallon ; 15 gallons at 14*. Qd. a gallon ; and 18 gallons at 15*. M. a gallon : what will be the price of a gallon of the mixture, so that he may gaiu £2. 5*. Gd. on his outlay ? MISCELLANEOUS QUESTIONS AND EXAMPLES. 133 (0) A gentleman distributed £41. 5s. among 12 men, 16 women, and 30 children; to every man he gave twice as n:iuch as to a woman, and to every woman three times as much as to a child : what did each receive ? (7) A chain, 11 yards long, is divided into 50 equal parts, called links; find how many square links there are in an acre. (8) A merchant expends £1036. 5s. on equal quantities of wheat at £2. 2s. a quarter, barley at £1. 1^. a quarter, and oats at l-is. a quarter : what quantity of each will he have 1 VII. (1) How many minutes are there in the 10 years, of which the first is 1852 ? (2) Divide 425 tons, 15 cwt., 2 qrs., 12 lbs., by 27: and 1361 m., 4 fur., 28 po., by 28 : and find how many moidorcs are equivalent to 198 guineas. (3) Two boys run a race of 1 mile, one of them gains 5 feet in every 110 yards ; how far will the other be left behind at the end of the race ? (4) Light travels at the rate of 192000 miles a second: how many days will it be in coming to us from the star a Centauri, supposed to be 20 billions of miles distant ? {5) Divide £100. 2*. Gd. equally among 45 people ; supposing 20 of them to have received their portions, and 10 of the remaining 25 to have given up their portions to the other 15, how much would each of the 1 5 receive ? (6) A father left his eldest son £5000 more than he left his second son, and the second son 1500 guineas more than the third ; to the third he left 12000 guineas : what was the eldest son's portion; and what sum did the father leave to his 8 sons? (7) A person buys 128 gallons of wine at Ss. Gd. a gallon : how many gallons of water must be added to it, in order that he may gain £5 12.y. on his outlay, and retail the wine at 5s. a gallon ? (8) A bankrupt has good debts to the amount of £456. 18^. 8d. ; and the following bad debts, £360. 7*. Gd., £120. IS^., and £21. 4^., for which he receives respectively 4, 5, and 10 shillings in the £ ; his own liabilities amount to £4558 ; how much can he pay in the £ ? VIIL (1) Can you attach any meaning (1) to the multiplication of Gs. Sd. by £1. 2.9. 3d, (2) to the division of 1 yard, 2 feet, 3 inches, by G feet, 8 inches? State reasons for your answer. 134 ARITHMETIC. (2) A carriage-load is found to weigh 1 ton, 3 cwt., 1 qi*., and it consists of 815 equal packages ; what is the weight of each? (3) A gives to B 98 gallons of brandy worth 255. 6d. a gallon, and gets in return 39 guineas and 570 yards of cloth : what is the value of the cloth per yard ? (4) A person counts on the average 7000 shillings in an hour : what sum will he ^ount in 67 days, if he work 9 hours a day ? (5) A gentleman's average daily expenditure for the year 1852 is £2. Os. lid. ; and this allows him to lay by £50 at the end of the year: what is his income ? (6) Shew how to perform the following operations: (1) the addition of £896. 5 fl. 4 c. 7 m. ; £301. 5 il. 8 c. 8 m. ; £23. 9 c. 6 m.: (2) the sub- traction of the second sum from the first; and (3) the multix^licationof the third by 248 ; reading off each result. (7) A grazier left to his 5 children in equal partions 175 oxen, 2003 sheep, 663 pigs, and 87 fowls : what was the value of each of their for- tunes, supposing the oxen to be worth 11 guineas each, the sheep a guinea and a half each, the pigs half-a-guinea each, and the fowls dd. each ? (8) I hire a house at £90 a year ; which is assessed in the rate-book at fths of its rent ; I agree to pay the rates upon it, viz., 3 poor's-rates of 9d., lOd., and l6\ 2d. respectively in the £, a church-rate of 8c?. in the £, and a paving-rate of 1*. 7d. in the £ : What is the whole annual cost of the house ? IX. (1) Explain the calendar as now in use. On June 21 of 1851 the Duke of Wellington had lived 80,000 days. Find the day and year of his birth. (2) The fore-wheel of a carriage is 10 feet in circumference, and the hind-wheel is 16 feet: how many revolutions will one make more than the other in 100 .miles? (3) A loaded truck weighs 4 tons, 3 qrs., 1 lb. ; the truck itself weighs a ton and a half, and it contains 758 equal packages : find the weight of each package. (4) A has 35 ponies, each worth 15 guineas, and B has 24 horses, each worth £24. 15*. : should they exchange, which of them ought to give money also, and how much ? (5) Sound travels at the rate of 1142 feet a second: if a gun be dis- charged at the distance of 4^ miles, how long will it be, after seeing the flash, before I hear the report ? MISCELLANEOUS QUESTIONS AND EXAMPLES. l^Ss (C) How many times \Yill a clock, which clumes the quarters, strike and chime in 1854 ? (7) How long will a person be in walking from Cambridge to Ely, a distance of IG miles, when he takes 110 steps of 2^ feet every minute ? (8) A manufacturer employs GO men and 45 boys, who respectively work 10 and 14 hours per day durmg 5 days of the week, and half the time on the remaining day ; each man receives Gd. per hour, and each boy 2d. per hour : what is the amount of wages paid in the year ? (a year = 52 weeks). X. (1) What will be the expence of forming a railway 14G miles in length, at 3 guineas a yard 1 (2) A gentleman's income is 2000 guineas ; ho spends £18 per week upon personal expences, and his annual subscription to charities amounts to £150 ; what will be the state of his finances at the end of 8 years ? (reckoning 52 weeks to the year). (3) Find the value of 12 lbs., 8 oz. of copper coin, having giveia that 12 penny pieces weigh 8 oz. (4) What is the price of 7 packages of cloth, each package contain- ing 7 parcels, each parcel 27 pieces, and each piece 81 yards, at the rate of Ih guineas for 3 yards t (5) A mixture is made of 6 gallons of spirits at 6 fl. 2 c. 5 m. per gallon, 4 gallons at 9 fl. per gallon, and 10 gallons at £1. 1 flc 1 c. 5 m. per gallon ; find the price of a gallon of the mixture. (G) If 5000 people took in hand to count a billion of sovereigns, and beginning their work at the commencement of the year 1852, could each count on the average 100 sovereigns a minute (without intermission), when would they finish their task ? (7) I have a bank-note of £20, a note-of-hand for £6. 10*. and in several coins, as follows ; in copper, 13 farthings, and 45 half-pence ; in silver, 86 three-pences, 58 groats, 96 sixpences, 67 shillings, 97 half- crowns, and 126 crowns; in gold, 65 half-guineas, 77 guineas, and 84 moidores: how much have I altogether? (8) In a manufactory there are employed 5 foremen, each at 4^. (kU a day, 63 workmen, each at 2^. 9 -t '■^-^ • / / . - ^>- Q« "r."'- therefore required value =2 pks.. If qts. — 5f qts. "_^ ^ «« -« .C- = 1 pk., 4/5 qts. I'f i y E ii '-« = 140 ARITHMETIC. Ex. XLVIII. (1) Find the respective values of 1 . ^- of £1 ; § of £1 ; | of £1 ; f of a guinea ; | of a guinea. 2. ■{- of £1. 1*05. ; I of £2 ; § of half a-crown ; f of 13*. 4d. 3. ^VV of ^'1 ; 1% of Is. ; I of 6s. Qd. ; § of 1*. Gd. ; ^ of 3*. ed. 4. 2i of 7*. ed. ■ f of £2. 8s. 9d. ; ^^ of a moidore ; § of £135. 16s. lOld. 5. 2 of 4s. Id. ; 1 1 of £1 . 2s. 9J. ; y\ x fi- of 21*. ; l of f of 9*. lO^d G. SjV of 2s. 6d. ; 3^5- of £4. 14*. 6d. ;" | of ^ of 10*. 6(/. ; ^ of 100 guineas. 7. f of f of |0- of 5 guineas ; f of £16. 16*. S^d ; y^oo of £441. 12.9. 6^." 8. •j^g- of a cwt. ; f of a lb. Avoird. ; f of a mile ; f of an acre. 9. i\- of a mile ; -f'^ of a day ; | of a yard ; f of 8 cwt., 1 qr., 14 lb. 10. 7| of a lb. Avoiid. ; 1| of a lb. Troy ; 2f of a gal. ; 4/^ of an acre. ] 1. 8j4j- of a hhd. of beer ; 2| of a tun of wine ; 6|f of a bus. 12. 2} of a load ; 8^ of a cub. yd. ; 9i% guineas. 13. I of f of lOf hrs. ; £ ^ ; |f of ^f of a moidore. it ^r 7-5 14. -j^^yi of £16. 8*. Ud ; f of If of 121 of ^ of £2 x ^. 15. f of £1 X 5f ; I of f of £1 -^ J. 16. 19| of £5. 1*. lid. ^q. ; 2t of £8. 14*. 23^.-=- 84. (2) Find the values of 1 . f of £1 + 1 of a guinea + 3*. 2d. 2. fof £l+|of2*. 6rf. + |ofl*. 3. ■^of£l + |of l*. + /^rf. 4- 2% of \l of 1 0*. 6d. + 1*. + yV of 2*. 6d. 5. I of £1 — I of Is. + 1 of a guinea— f of a moidore. 6. £S^^+7ts. + ^d. 7. f of £1 - f of 2*. 6d. + f of 1*. 8. I of 10*. 6d. + 1 of 27*. - A of 6s. Sd. 9. ^ of £1 . 1 25. + ^5 of £3. 5s. + ^-^ of 1 ^ guineas. 10. f of I of £1 + 1 of I of 2s. 6d. + J of 10^-rf. 11. f of 21*. + ^ off of £1-^ of I of 5*. + ^ of I of U 12. ^of£15 + |of i^ of£1.12*. + f ofSd ^6 REDUCTION OF FRACTIONS. 141 13. ;^ of -j^ of 2 guineas + i|^ of £5. 14. f ofaton + f ofacwt. + f lb. 15. -I lb. Troy + f lb. Troy - f oz. Troy. 16. 1^ of a mile — | of a fur. + j^jpo. 17. Yiy cub. yds. + 2} cub. ft. 18. § of a qr. + f of a bus. — ^ of a qr. 19. I of 7 fur., 29 po., 8} yds. + f of 5 mi., 3 fur., 37 po., 4^ yds. 20. 7f of 3651- d. + 3i»5 of | wks. + 1 of 5f hrs. 21. -^j of 91 ac, 3 ro., 8G po., 2| yds.-| of G ac, 2 ro., 17 po. 25 J yds. 133. To reduce a numheVj or afractiouj of any denomination, to afrao^ tion of another denomination. Rule. " Reduce the given number, or fraction, and also the number or fraction to the fraction of which it is to be reduced, to their respective equivalent values in terms of some one and the same denomination : then the fraction of which the former is made the numerator^ and the latter the denominator, will be the fraction required." Ex. 1. Reduce 8*. 5c?. to the fraction of £\. Proceeding by the above Rule, 8*. bd. = 41 pence, £1 = 240 pence ; therefore fraction required = ^^^. Reason for the above process. £1, or unity, is here divided into 240 equal parts ; and 41 of such parts being taken, the part of unity, or £1, which they make up, is represented Ex. 2. Reduce | of £1 to the fraction of 27*. |- of £1 = 20 times I of 1*. 5x20 6x6 = — ^• 27s. = 27s. 5x5 2 therefore fraction required 6x_5 1 _25 2 "" 27 54 142 ARITHMETIC. For 27*. is divided into 27 equal parts ; and -^ of £1 is divided into ^^ of such parts ; therefore the part of unity, or 27s., which the latter repro- , . ¥ 25 sents^is^-^^^-. Ex. 3. Express f of £1 as the fraction of a farthing. f^ of £1 = (f X 20 X 12 X 4) farthings, = 5JLSi) farthmgs ; 1 farthing = 1 farthing ; therefore fraction required = -^ — , = ^~%^. For the unit, or farthing, is divided into 1 part, and £f contains ?-920 of such parts. Therefore the fraction of unity, or 1 farthing, which f of £1 representSj . i^^^o 1920 Ex. 4. \\'liat part of ^ of a ton is 2§ of 1 ^ of ^ of a cwt. ? 2| of 1 J of I of a cwt. = I of f of f of a cwt. = f xf x^cwt. J of a ton ="3^ cwt. •§- X ^ X J- Therefore fraction required = —^q — - Ex. XLIX- (1) Reduce 1. Gs. 8J. to the fraction of £1 ; and 3*. lid. to the fraction of 1 guinea. 2. 5d. to the fraction of Is. ; and 3s. 4^d. to the fraction of £1. 3. I^d. to the fraction of 27*. ; and 15 sixpences to the fraction of 135. M. 4. £1. 3*. M. to the fraction of £9. 6s. 8d. ; and 2^. Old. to the fraction of 10^. Gd. 5. £i. I7s. Gd. to the fraction of £5; and 16*. to the fraction of £200. e. £18. 7*. Gd. to the fraction of £2; and 6s. 7^d. to the fraction of 7*. 9c?. REDUCTION OF FRACTIONS. 143 7. 1^. 2d. to the fraction of a moidore ; and 3s. M. to the fraction of a half-guinea. 8. 3qrs., 19 lbs. to the fraction of a ton; and 61| lbs. to the frac- tion of 4 oz. 9. 3 qrs., 4 lbs. to the fraction of 2 cwt. ; and 5 oz., 2| drs. id the fraction of a grain. 10. 8 ro., 27^ po. to the fraction of an acre ; and 26| sq. yds. to the fraction of 2 acres. 11. 126yds., 2ft., Gin. to the fraction of a mile; and 6 cub. ft., 100 cub. in. to the fraction of a cubic yard. 12. 2 qrs., 2s na. to the fraction of an Eng. ell ; and 8 h., 3 m. to the fraction of a day. 13. 1 stone, 8 lbs. to the fraction of a ton ; and 1 sc, 13grs. to the fraction of a lb. 14. 2ac., 1 ro. to the fraction of 9ac., 2 ro. ; and 1540 yds., 2ft., 9 in. to the fraction of 2 miles. 16. 1 ft, I in. to the fraction of a sq. yd. ; and 2 qts., Ih pt. to tho fraction of a barrel. 16. 2 wlvs., 5 days, 7 h., 27 m. to the fraction of a day ; and 1 ro., 20 po. to the fraction of an acre. 17. 4 bush., 2| qts. to the fraction of a load ; and 3 quires, 7 sheets to the fraction of a ream. 18. 2| guineas to the fraction of £2\; and 2|-cwt. to the fraction of 2 tons, 12 lbs. 19. lOf months to the fraction of 13 months ; and 100^ guineas to the fraction of a groat. 20. 6 ft., 8| in. to the fraction of 13 ft., 8^^ in. ; and 1.^ yds. to the fraction of 1| in. (2) Reduce 1. f of a crown to the fraction of £1 ; and f of a farthing to the fraction of 1*. 2. f of Is. to the fraction of a guinea ; and i of 7*. to the fraction of a crown. 3. f of a g-uinea to the fraction of £\ ; and |- of 27*. to the fraction of 25. Qd. 4. ^ of a half-guinea to the fraction of £1 ; and j# of Is. to the fraction of 2*. Gd. 5. I of £74. 13s. 4:d. to the fraction of X28 ; and ^ of a moidore io the fraction of 3| guineas. 1 44 ARITHMETIC. 6. I of a dwt. to the fraction of 1 lb. ; and | of 2 lbs. to the frac- tion of 2^ tons. 7. I of a lb. to the fraction of a cwt. ; and | of a yd. to the frac- tion of a mile. S* 3otr o^ ^1 *o ^^^^ fraction of a penny; and ^^ of a mile to the fraction of U yard. 9. f of 4" of half-a-guinea to the fraction of 2*. 6d. ; and 1 oz. Troy to the fraction of 1 oz. Avoirdupois. 10. f of a pole to the fraction of a league ; and 8| furlongs to the fraction of 2J miles. 11. I of 7-^ of 161 yards to the fraction of a furlong ; and ^ of ^ of a guinea to the fraction of 2.?. 6d. 12. ^ of IGs. Ohd. to the fraction of 17*. Gd. ; and ^[^ of a lb. Troy to the fraction of a pennyweight, IS. I of a lb. Avoird. to the fraction of 2 lbs. Troy ; and |- of 2*. Gd, to the fraction of 1 1 guineas. 14. f of a French ell to the fraction of a yd. ; and | of a crown to the fraction of ^ of 7*. Gd. 15. 1^ of a sq. in. to the fraction of a sq. yd, ; and ^ of a yd. to the fraction of an English ell. 16. ybosu of ^ y^^^ *® *^^ fraction of a day ; ^-'tt^^ to the fraction of a farthing. (3) 1 . What part of 7 guineas is -| of a moidore ? 2. What part of £9 is ^ of ^^ of half-a-crown ? 8. What part of a second is yoWchj ^^ ^ '^''^y '• 4. What part of f of a league is ^ of a mile ? 5 What part of 4^ guineas is 5| of jSr of £7 ? 6. What part of 3 weeks, 4 days, is i- of 5^ sec. ? 7. What part of ^ of an acre is 25 ^j po. 1 8. What part of yjij of a min. is j^ of a month of 28 days? 9. What part of ^ of 4 tuns of wine is 8} lihds. ? 10. What part of 3 fathoms is -^-^ of ^ of a pole ? Examples, such as the following, are often given, Ex. 1. Compare the values of rr^ of £1, 2ij of ^ guinea, and J of 3«. Hd. knJ-^d.^^^^ 21 KEDUCTION OF FRACTIONS. 14j 1 r • 21x12, -.,., v^.T of a guinea = — — — a. = \-f a., I of 3s. m. - i of 45 J-(/. = ( J X y-)d. = V ^. Therefore the equivalent fractions in one and the same denomination (namely, that of pence) are respectively ^, \\«, and V. Least common multiple of the denominators = 7 x 11 x 8 ; therefore tlxC fractions become respectively 80 X 11 X 8 7040 7x 11 x8 ■ 616 * 126 x7 x8 7056 11 x7 x8 ■616 ' 91 X 11 x7 7007 8x 11 x7 ■ 616 ' therefore ^ of a guinea is the greatest, -^ of £1 is the next, and ^ of O.S. 9hd. is the least. Ex. 2. Express £^— f of a guinea as the fraction of half-a-erown. 9x20 5x21 ^T^-f of guinea: 10 6 -(9x2>.--^5. Half-a-crown = 2|;?.=f*. ; therefore the fraction required =§- = |. Ex. 8. Reduce ^- of \^ of £1-^ of Is.} to the fraction of A moidore. 10 146 ARITHMETIC. 26 ^( 19x8 7) 9 ""MexS "48) (26 145 \ A moidore = 27^. ; 26 145 9 ^ 48 therefore the fraction required = ' ^^ 13x145 ~9x 24x27 _ 1885 ^5832* Ex. 4. What fraction of a guinea together with 4*. 6d. is equivalent to 15*.? In other words the question is, " What fraction of a guinea is equiva- lent to 155. -4*. 6d., or 10*. 6d. V Now 10*. Qd. = 21 sixpences, 1 guinea = 42 sixpences; therefore fraction required = |i = l. Ex. L. (1) Compare the values of x\ of £1, 5^ of a guinea, and ^ of a crown. (2) Compare the values of f of £1, f of a guinea, and § of 15*. 7hd. (3) Which is the greater, ^^ of a day, or f of an hour, and by how much ? (4) Express the difference between J^ of £1 and ^^ of a guinea as the fraction of half-a-crown. (5) Express the difference between f of a guinea and ^ of £1 aa the fraction of half-a-guinea. (6) Reduce | of a crown +f of a shilhng to the fraction of a guinea. (7) Express § of 2*. Gd. + 1 of a gumea + f of £1 - ^^ of a penny as the fraction of £5. (8) Add together f of £3. 7*. 6d. and f of J of 4J- guineas ; and reduce the result to the fraction of £2. REDUCTION OF DECIMALS. 147 (0) What fraction of £10 together with 3| guineas is equivalent to 5 guineas ? (10) "What fraction of 2^ cwt. together with 3 qrs., 14 lbs. will give a ton and a half? REDUCTION OF DECIMALS. 184. To reduce a decimal of any denomination to its proper value. Rule. " Multiply the decimal by the number of units connecting tho next lower denomination with the given one, and point off for decimals as many figures in the product, beginning from the right hand, as there are figures in the given decimal. Tlie figures on the left of the decimal point will represent the whole numbers in the next denomination. Pro- ceed in the same way with the decimal part for that denomination, and 50 on." Ex. 1. Find the value of '0484 of £1. Proceeding by tJie Rule given above, •0484 20 •9680^. 12_ ll-eiGOd. 4 2-4G40g. therefore the value of -0484 of £1 = Ud. 2-4640^. = Ud.2^^^q. Reason for the above process. 9680 1161G0 , do 10000 • 10000 'lOOO 11 fiifi ^ 11 J , 616x4 = "^-^ioob^- = lld+2^g, 10-2 148 ARITHMETIO. Ex. 2. Find the value of 18-8375 acres. Acres. 13-3375 4 1-3500 1-0. 40 14-0000 po. therefore the value is 13 ac., 1 ro., 14 po. Kx. 8. Find the value of '07 of £2. 10*. £2. 105. = 506-. •07 60 8-50*. 12 6'OOd. therefore the value of -07 of £2. 10s. is Ss. 6d. Ex. 4. Find the value of -0474609875 of £10. 13*. ^dc £10.1Ss.M. = 25G0d. •0474609375 2560 28476562500 2373046875 949218750 121-5000000000^. 4 2-Og. therefore the value is 121|d or 10s. l^d. Ex. 6. Find the value of •972916 of £1, £. 1st method. -972917 20 19'4583405. 12 5-500080d 4 2-000320g. therefore the value is 195. d^d. neai-ly. REDUCTION OP DECIMALS. 149 2nd method. •072910 of £^= ^™^~J™^ of £1, Art. (97), 875625 900000 --K:io-«)^ 467 24 N'ote. The latter is generally the better course to adopt. Ex. 6. Find the value of '375 of a guinea + '54 of 86-. Sd. +-027 of £2. Us. Guinea. •375 21 875 750 7-875,9. 12 lO-SOOcf. 4 2'000^, therefore *o75 of a guinea = 7-!^'. 101^. •54 of 8*. 8c?. = ^ of 99c?. = 54c?. = 4?. 6fi. •02f of £2. 15s. = (^ of 55^ s. therefore the value required = 7^. lOhd. + As. Qd. + ls. 6d, = 13?. lO^rf. Ex. 7. Find the value of ^— of 82- tons — '8405 of 1§ qrs. + .^ of2cwt., 102 lbs. — of 83 tons „, A33 15\, 183x3, ftons^-^— x-jtons=-g^^tons, /1 38x3 ^ \ , 399 ^ \ 80 X 4 / lb =24 cwt., 3qrs., 211bs. 150 ARITHMETIC. .oin- ri2 /S405-3 „5\ •2IS348 ._ , ,^„., /218348-218S4 1000 . ^^^N ,, of2cwt,1021bB.= ^__^^^^^^_x-^of 326jlb3. ■lbs. •S26 _ 192014, 900" = 213 1'"- therefore the value of the expression = 24cwt., 3 qrs., 21 lbs.-15|f lbs. + 213i|^lbs. = 24cwt., 8 qrs., S^^lbs. + l cwt, 3 qrs., I71|-^lbs. Ex. LI. (1) Find the respective values of 1. -45 of £1 ; -10875 of £1 ; 87708 of £1. 2. -28126 of £1 ; -7962 of £1 ; •859375 of £2. 3. 600625 of £1 ; -775625 of £5 ; •0875 of 16*. . 4. -0625 of a guinea; -7635 of 105. ; 2 625 of 1.?. 6. -056713 of a guinea ; 2 76548 of £1 ; 1-74875 of 10*. 6. 3049 of £1 ; 0425 of £100 ; -432 of 13*. 4i. 7. '1875 of 5 guineas ; 1-05625 of 6s. Sd. ; -875 of £3. 6*. Gd. 8. 810532 of 12s. Gd. ; 275 of 2*. 4d. ; 41-375 of 8d. 9. -875 of a lea. ; 2 ■5884375 of a day ; "6 of 1 lb. Troy. 10. 6156510416 of £4 ; ^046875 of 3 qrs., 12 lbs. 11. -85076 of a cwt. ; 07325 of a cwt. ; '045 of a mile. 12. 4'16525 of a ton ; 3-625 of a cwt. ; 05 of an acre. 13. 3-8343of alb. Troy; 2-46875 ofaqr.;4-100of 3cwt.,l qr.,211ba. 14. 3-8875 of an acre ; 3-5 of 18 gallons. 15. -925 of a furlong ; •34875 of a lunar month. 16. 6-06325 of £100 ; 3-8 of an Eng. ell. 17. 2-25 of 31- acres; 2-465 of 5 crowns. 18. 1-605 of £3. 2*. 6d. ; 2-0396 of 1 m., 530 yds. 19. 4-751 of 2 sq. yds., 7 sq. ft. ; 2 0005 of £63. 0*. S\d. 20. 2-009943 of 2 miles ; 1-005 of 15 guineas. REDUCTION OP DECIMALS^ 151 (2) Find the respective values of 1. -383 of £1 ; -47083' of £1 ; -4694 of 1 lb. Troy. 2. -5746 of 275. ; -188 of 105-. Gd. ; 2 G of 5s. 3. -142857 of 2 guineas ; 3-2095328 of 17^. 6i. 4. -0G3 of 100 guineas ; 2 0138 of S'S moidores. 5. 405 of Uj; sq. yds. ; -163 of 21 miles ; 4-9d of 4 d., 3 his. 6. 8-242 of 21 acres ; ^~ of 2.V of 2*5 days. -5681 (3) Find the difference between '77777 of a pound and 8^. QGQiSd. ; and between •70823 of a pound and 3 5646 of a shilling. (4) Subtract | of a crown from £1-59375. (5) Find the respective values of the following expressions: 1. -68125 of £1 + -875 of 135, 4i. + -605 of £3. 2s. Qd, 2. 31 of 6s. 8d. - -40972 of a guinea + 2-75 of £30. . 3. £ 634375 + -025 of 25*. + -316 of SO^. 4. -75 of 6s. 8d. - 1 -84375 of 4s. + 3-9796 of 2s. 6. 2-81 of 8651 days + 5-75 of a week - 1 of 5| hours. G. f of j\ of 3 acres -200875 square yards + '0227 of 3| square feet. (6) Which is the greater, -0231 of a guinea, or -19 of a half-crown? 135. To reduce a niimher or fraction of any denominatioiij to the decimal of another denomination. Rule. " Reduce the given number or fraction, to a fraction of the proposed denomination ; and then reduce this fraction to its equivalent decimal." Ex. 1. Reduce | of £1 to the decimal of 1 guinea. 1 guinea =21 5. therefore the fraction required = 7^^. 4 80 1-14285714 7 is _ •38095238 therefore the decimal required = -38095238. 152 ARITHMETIC. Ex. 2. Reduce 13^. G^d to the decimal of £1. therefore the fraction = £1 = MOd, ; 649 4_ 649 240 ~ 960* 960) 649 00 (-67 5760 7300 6720 680 We may work such an example as the ahove more expeditiously, hy first reducing ^d. to the decimal of a penny, which decimal will be '25, and then reducing 6*25d to the decimal of a shilling hy dividing by 12, which decimal will be -520833, and then reducing 13-520833^. to the decimal of £1 by dividing by 20, which process gives '67604166 as the required decimal of £1. The mode of operation may be shewn thus : 4 1-00 12 6-25 2,0 13-520833 •6760416 Ex. 5. Reduce S bus., 1 pk. to tho decimal of a load : and verify the result. 1-00 8-25 •40G25 •08125 therefore '08125 is the decimal required. '08125 Id. •40625 qrs. 8 8-25000 bush. 4 1-00000 pk. therefore •08125 of a load = 8 bus., 1 pk. REDUCTION OF DECIMALS. 153 Ex. 4. Add together f of 21*., J of a moidore, f of 7*. Qd., and reduce the result to the decimal of £1. ^oi21s. = ^8. = Ss.4^. ^of27s. = ^^s. = £l.0s.8d. rf of 7s. Gd. = (3s. 9(/. ) X 5 = 1 Qs. dd. therefore the sum = £2. 75. 4f f/. Now to reduce £2. 7-9. ^d. to the decimal of £] . 5 4- 12 48 2,0 7-4 •37 therefore the decimal required = 2"37. Ex. 5. Express the sum of '428571 of £15, I of ^ of f of £1. 125., and f of 3d., as the decimal of £10. •428571 of £15 -Iff .^ of £15. = fof£15 = £V = £G. 8*. Gf^.j 1 of ^ of f of £1. 12.. = i of ^\ off of 32.. = Ys. = 2s. 3fd; foi3d. = \fd.^l^d.; therefore the sum = £6. Ss. 6^d. + 2s. 3fd + lfd = 6.11.9. 2,0 11 10 6-55 •655 therefore the decimal required = "655. Ex. 6. Convert £17. 9*. 6d. into pounds, florins, &c. ; and verify the result. First reduce 9^. Qd. to the decimal of £1. 12 2,0 ~ -475 GO 9-5 154 ARITHMETIC. .-. £17. 9s. 6d. = £17-475 =£17. 4fl. 7 c. 5in. Again, £17. 4 fl. 7 c 5 m. = £17-475 20 9-500*. 12 6000 d .-. £17. 4fl. 7 c. 5 m. = £l7. 9s. 6d, Ex. 7. Express 1 shilling and 1 half-crown in terms of the decimal coinage. 1*. = £^Q = £j-^ = £-05 = 5 cents ; 2c?.6a?.=-£i = £JJ^=£-125 = lfl. 2 c. 5 m. Ex. 8. Reduce the difference between a cent and a penny to the decimal of 2s. 4d. -^d. = £^', lc. = £x^; .-. difference = £(iJ^-^) = £2i-^^ = £j^ = (j^x20xl2)d. = ld. Ss.4.d. = 40d. z ••• fraction=:A=.^_7_^_^g^. .*. decimal =035. Ex. Lll. (1) Reduce 1. Gs. M. to the decimal of £1 ; and 8s. BM. to the decimal of £1. 2. 4s. 7ld. to the decimal of £1 ; and 155. ll\d. to the decimal of £1. 3. 3*. 4-?fC?. to the decimal of a crown; and |d to the decimal of£i. 4. 10*. 02 c?. to the decimal of £1 ; and 5*. 8|d to the decimal of £5 5. 1*. o\d. to the decimal of 15*. ; and 12*. l\d. to the decimal of a guinea. 6. 5*. to the decimal of 13s\ 4d. ; and 1 8*. 9d. to the decimal of 27*'. 7. 13*. Qd. to the decimal of 10*. ; and £1. 9*. 4|d to the decimal of£l. REDUCTION OF DECIMALS. 155 8. £3. lis. 0^. to the decimal of £1 ; and also to the decimal of £2. 10s. 9. 14^. O^d. to the decimal of 3 guineas ; and 27^. to the decimal of 1^ guineas. 10. 6.V guineas to the decimal of £5 ; and V.d. to the decimal of £100. 11. £8. Os. lOd. to the decimal of 5hd. ; and 7 guineas to the deci- mal of £5. 105. 11(/. 12. 2oz., 13dwts. to the decimal of 1 lb.; and 41bsi, 2sc. to the decimal of 1 oz. 13. 2 qrs., 21 lbs. to the decimal of 1 ton ; and 3 cwt., 3 oz. to the decimal of lOcwt. 14. 2 fur., 41 yds. to the decimal of a mile ; and 1 fur., 80 po. to the decimal of a league. 15. 2sq. ft., 73 sq. in. to the decimal of a square yard ; and 3ro., 20 po. to the decimal of an acre. 16. 14 gals., 2 qts. to the decimal of a barrel ; and 3 qrs., Spks. to the decimal of a load. 17. 4 days, 18 hrs. to the decimal of a week ; and 11 sec. to the decimal of 5 days. 18. 1^ guineas to the decimal of £1|; and 1 lb. Troy to the deci- mal of 1 lb. Avoirdupois. 19. 2} inches to the decimal of 2^ miles; and 1 st, G^lbs. to the decimal of 8| lbs. 20. SJpks. to the decimal of 3^ qrs.; and 27i^gals. to the decimal of n qts. 21. 5|yds. to the decimal of 2 Fr. ells ; and 1 ton, 2^- cwt. to the decimal of 1 cwt., 2} qrs. 22. 3wks., 5id. to the decimal of 5^ hrs. ; and 1 min., 2Jsec. to the decimal of ^ of a lunar month. 23. 3 reams to the decimal of 19 sheets; and 3^ acres to the decimal of 3j sq. yards. 24. 83 yds. to the decimal of a mile ; 3^. 5^^c?. to the decimal of a dollar, a dollar being 4*. 8d.; and 7*. ^ j^oo^a d. to the decimal of 105. Gd, (2) Reduce 1. I of 135. Gd. to the decimal of £1 ; and ^ of half-a-cro\\'n to the, decimal of l5. 2. f of a crown to the decimal of 21 s. ; and 65 cwt. to the decimal of a ton. 156 ARITHMETIC. S. ^ of a guinea to the decimal of £1 ; and | pk. to tlie dec! ma- of 2 qrs. 4. f of a guinea to the decimal of £2 ; and -j-f^ of a year to the decimal of a day. 5. ^ of Jjj of 40 yds. to the decimal of ^ of 2 miles ; and i of 8'}sq. yds. to the decimal of 2 acres, 1 ro. 6. f of 4| hrs. to the decimal of 8651 days ; and 9^j of H pecks to the decimal of 3^ qrs. 7. 3 lbs., 6oz. Troy to the decimal of 10 lbs. Avoird.; and i oz. Avoird. to the decimal of i oz. Troy. (3) Express | of a cro^vn + f of a shilling as a decimal of 7^. (4) Express | of half-a-cro wn + '4 of a shilling as a decimal of £2. (5) Add together | of a day, | of an hour, and f of 6 hours ; and ex- press the result as the decimal of a week. (6) Express the difference of f of a guinea and £- of 7^^. 6d. as the decimal of a moidore. (7) Express the value of '83 of 8s. + "05 of 2 guineas + 1*8 of 5s. as tlio decimal of half-a-guinea. (8) Find the difference between 6i half-guineas and £3"525 ; and reduce the result to the decimal of a crown. (9) Add 5|cwt, to 3"125 qrs.; and reduce the sum to the decimal of a ton. (10) Convert the following sums of money into the decimal coinage of pounds, florins, &c., and verify each result : 1. 6d. 2. 10c?. 8. Ud. 4. 5.?. 5. 10s. ed. 6. 16^. ^ 7. £5.12s.6d. 8. >£54:.7s.4:d. 9. £20. Ids. 7},d. 10. 15*. 43^ 11. 145. 816d 12. £2. 15s. IVOndd. 13. £8. Os. Ud. S'Oiq. PRACTICE. loG. Def. An Aliquot Part of a number is such a part as, when taken a certain number of times, will exactly make up that number. Thus, 4 is an aliquot part of 12, 6 of 18, &c. Practice is a compendious mode of finding the value of any number of articles by means of Aliquot Parts, when the value of an unit of any jdenomination is given. Practice may be separated into two cases, Simple and Compound. I. Simple Practice. In this case the given number is expressed in the same denomination PRACTICE. 157 as the unit whose value is given : as, for instance, 26 lbs. at £2. 5s. per lb.; or 830 articles at 5s. G^d. each. The Rule for Simple Practice wiil be easily shewn by the following examples. Ex. 1. Find the value of 129G things at IQs. lO^d. each. The method of working such an example is the following : Supposing the cost of the things to be £1 each ; then the total cost = £1296; therefore £. cost at 106\ Od. each = J of the above sum = 648 cost at 5s. Od. each = ^ the cost at 106\ each =32-i cost at 1^. 3c?. each = ^ the cost at 5s. each ... — 81 cost at Q.y. 7ld. each = l the cost at 1^. Qd. each= 40 therefore, by adding up the vertical columns, s. d. 10 cost at 16*. lO^d. = £1093 . 10 . The operation is usually written thus : 105.=!- of £1 £. 1296 . s. d. = cost at £1 each. 5s. = loil0s. s.2d. = loi5s. 7^d. = }roils.Sd. 648 . 824 . 81 . 40 . 10 O = cost at 10*. each. . = cost at 5s. each. O = cost at 1*. 8d. each. . O = cost at7^d. each. £1093 10 . = cost at 16*. lOU. each. N'ote. The student must use his own judgment in selecting the most convenient 'aliquot* parts ; takmg care that the sum of those taken make up the given price of the unit. Ex. 2. Find the value of 8825 things at £2. 17*. 4W. each. £. s. d. = value at £1 each. 10*. = iof£L 5s. = }roil0s. 2s. = loil0s. (/.takeiof £1912. 10*.) 4d = lof2s. ld. = loUd. 3825 . . 2 7650 . 1912 10 , 956 5 882 10 63 15 . 7 19 . 4? £10972 . 19 . 4^ = value at £2 each. = value at 10*. each. = value at 5*. each. = value at 2*. each, = value at M. each. 4-^ = value at Id. each. value at £2. 17*. Ud. each. 158 ARITHMETIC. Ex. 3. Find the cost of 165| cwt. at £2. 5s. 6c?. per cwt. The cost clearly = 165 times £2. 5^. 6i. + | of £2. 5*. 6d, £. s. d. 5s. = loi£h 165 6d. = ^oi58. 330 41 4^ 375 I of £2. 5s. 6d.= 1 £377 Ex.4. Find the 0.0= cost at £1 each. 2_ . =:costat £2each. 5.0= cost at 5s. each. 2,6= cost at 6d. each. 7 . 6 =cost of 165 cwt. at £2. 5s. 6d. per cwt. 19 . 9f = cost of I cwt. at £2. 5s. ed. per cwt. 5*. 6d. per cwt. 3f = cost of 165|cwt. at £2. value of 6413 things at 4^. lO^d. each. £. s. d. 4^. = iof£l 6413 6d. = ^oUs. 1282 12 . 4d. = ^o^4s. 160 6 . 6 ^d. = ^oMd. 106 17 . 8 11 . 13 . 91^ = value at £1 each. = value at 45. each, = value at 6d. each. = value at 4d. each. = value at ^^g^o?. each. £1561 . 9 . 11 Ji = value at 4s, lO^^d, each. II. Compound Practice. In this case the given number is not wholly expressed in the same denomination as the unit whose value is given ; as for instance, 1 cwt., 2 qrs., 14 lbs. at £2. 2s. per cwt. The Rule for Compound Practice will be easily shewn by the follow- ing examples. Ex. 1. Find the value of 84 cwt., 3 qrs., 14 lbs. of sugar at £12. 11*. Si. per cwt. The method of working such an example is the following : The value of 1 cwt. of sugar being £12. 11*. Sd., £. the value of 84 cwt. of sugar = 3057 2 qrs. = } (value of 1- cwt.) = 6 1 qr. = |- (value of 2 qrs.) = 3 14 lbs. = -^- (value of 1 qr.) = 1 therefore, by adding the vertical columns, 5 2 11 d. 10 11 the value of 84 cwt., 3 qrs., 14ib3.=:£1068 .0.2^ PRACTICE. 159 The operation is usually written thus : £. s. d. 2 qrs. = ^cwt. 1 qr. =^ of 2 qrs. 14ibs. = ^oflqr. 12 . 11. 8 12 151 . . 7 1057 . 6 . 3 . 1 . . 5 . 10 2 . 11 11 . 51 = value of 1 cwt. = value of 12 cwt. = value of (12 X 7) or 84 cwt = value of 2 qrs. = value of 1 qr. = value of 14 lbs. £1068. 0. 2| = valu3 of 84cwt., Sqrs., 141b3. Ex. 2. Find the value of 319 cwt., 3 qrs., 16 lbs. at £2. \1s. Qd. per cwt. £. s d. 2 qrs. = I cwt. sul)lracting 1 qr. = ^ of 2 qrs. 14lbs. = iof Iqr. 21bs.= jofl41bs. 2 . 12 . 6 10 26 . 5 . 4 105 . 8 840 . 2 . 12 . 6 887 1 7 6 13 6 = value of 1 cwt. = Vralue of 10 cwt. = value of 40 cwt. = value of 320 cwt. = value of 1 cwt. = value of 319 cwt. = value of 2 qrs. l?r = value of 1 qr. 6^ = value of 14 lbs. Ex.3, lft. = Joflyd . 111 = value of 2 lbs. £889 . 14 . 4^ rvalue of 819 cwt., 3 qrs., 16 lbs. Find the value of 37 yds., 2 ft., 7 in. of silk, at 6s. ^\d. a yard. £. s. d. 1ft. = Joflyd. 6 in. = |-oflft. lin. = ^ of 6 in. . 5 . 3i 4 1 . 1 . 1 9 9 . 9 . 9 9 . 5 . 8:1 15 1 1 0^ 9x\ lOif ~ value of 1 yd. = value of 4 yds. = value of 36 yds. = value of 1 yd. = value of 37 yds. = value of 1 ft. = value of 1 ft. = value of 6 in. = value of 1 in. £9.19. 6^d.%lq. = value of 37 yds., 2 ft., 7 in. 160 ARITHMETIC. Note. It will be found most convenient, in all examples of Practice, to work with fractions of a penny, and finally to find the value of the 3um of these fractions in farthings, as m the above example. Ex. LIII. Find the value of 1. G45 things at 25. Qd. each; and 69 things at 10*. ^d. each. 2. 454 things at 2^. M. each ; and 72 things at 1*. 7d each. 3. 62 things at 3*. 9c?. each ; and 1257 things at ^d. each. 4. 626 things at 8^. 8c?. each ; and 286 things at 125. \d. each. 5. 80 things at 4^. 4Jd each ; and 37 things at 5*. h\d. each. 6. 138 things at £1. 145. each; and 589 thmgs at £1. \\s. Gd. each. 7. 95 things at £1, 2s. 6d. each ; and 107 thmgs at £24. 6^. 2d, each. 8. 457 tilings at £1. 85. 6d. each ; and 88 things at 1^^. each. 9. Ill things at £2. 5s. lOd. each; and 9261 things at 14*. lid. each. 10. 4681 things at S^d. each; and 1209 things at 185. Id. each. 11. 1450 things at £1. 7*. Sd. each; and 249 thmgs at £2. 135. 9d, each. 12. 898 things at I85, 7fc?. each ; and 405 things at 195. S^, each. 13. 744 things at £19. 195. each ; and 421 thmga at £4. 25. 6^d, each. 14. 1593 things at 95. OU. each ; and 6602 thmgs at 75. If^. each, 15. 7382 things at£3.165.4i'ocess. 9ft. X 4 ft. = 36 sq.ft.; 9ft.xr=:=^9xl)sq.ft.= |sq.ft. 60 + 3\ -Uq.ft. G'x4ft. = ( -x4 = 5 sq. ft. + — sq. ft. i-Z ~ 5 sq, ft. + 3 supei-ficial primes. 24 sq. ft. = -- sq. ft. = 2 sq. ft. ; iZ = 3 superficial primes 4- 6 superficial seconds. Now 36 sq. ft. + 5 sq. ft. + 3 superficial primes + 2 sq. ft. + 8 superficial primes + 6 superficial seconds = 43 sq. ft. + 6 sup. primes 4- 6 sup. seconds -( = 434- 12x6 + 6' ■ j sq. ft. ]44 = 43 sq.ft. 78 sq. in. A'^ote. Attention to tlie accompanying geometrical figure may per- haps explain more clearly the A result obtained by multiplying ^^ 9 ft. by 7 primes. Take ^B = 9ft., ^C=7ft., AD = 7 in. Then 9 ft. x 7 ft., or rect. AB, AC = rectangular figure A CEB, ^vhich contains 63 sq. ft., and 9 ft. X 7 primes, or rect. AB, ^D = rectangular figure J2)jPJ5, C which is Jjth part of 63 sq. ft. For since there are 12 lines in AC, each =AD, it follows that there are 12 rectangular figures, each =ADFB in rcctan^^ular figure ACEB, SQUARE AND CUBIC MEASURE. 169 Ex. 2. Multiply 17 ft. 3 in. 6 pts. by 12 ft. G in. 3 pts. ft. 17 . 8' . r/ 12 . G . 3 207 . G . 8.7 . 9" . 0'" 4.3 .10'^'. e"" 216 . 6 . . l(y'' . &'" = 216sq.ft. + g+ ^-^ + j2^ + ^2x12x144) ^^•^^• ^21Gsq.ft.+ (^^^+^-^-f ^^^3^^^J sq.ft> „ ^ . 10 . 6 . = 21G sq. ft. + 72 sq. in. + -^ sq. in. + — sq. in. Ex. 3. Find by cross multiplication the capacity of a cube whose edgo is 2 it. 8 in. ; and prove the truth of the result by vulgar fractions. ft. 2 . 8 2 . 8 6 . 4 1 . 9 . 4" 7 . 1 . 4'/ 2 . 8 14 . 2 . 8 4 . 8 . 10 . 8^ as . 11 . G . 8 = 18cub.ft.+ (|i + A+_A)eub,ft. 1/ZO ^ISiffAcub. ft. = 18 cub. ft. 1664 cub. in. Proof by Vulgar Fractions. Content = (25 x 2f x 2|) cub. ft. /8 8 8\ , „, 512 , ^ = 18 cub. ft. 16C4 cub. in. 170 ARITHMETIC. 140. In the examples of Cross Multiplication we see that a mixed clecimal and duodecimal scale of notation is employed, the figures of the feet being expressed and multiplied in the ordinary \Yay; whereas in other places the number 12 is always used instead of 10 : Cross Multipli- cation is not, therefore, properly termed Duodecimal Multiplication or Duo'lecimalsj because, although the different denominations are con- nected with each other by the number 12, still the different digits of those denominations are connected with each other by the numbef 10. Ex. LIV. 1. Find the area of a rectangular board, whose sides are 2 ft. 9 in. and 10 ft. 4 in. respectively. 2. A room is 17 ft. Sin. long, and 13ft. 10 in, broad; find the area of the floor in feet and inches. o. Find the number of square feet and inches in a rectangular piece of ground 9 ft. 8 in. by 8 ft. 5 in. 4. The floor of a room, which is 15| ft. wide, contains 91 sq. yards ; find the length of the room. 5. A rectangular plot of ground 26 ft. broad contains 92 sq.yds. 4sq. ft.; find its length. 6. Find the breadth of a room, whose length is 22h ft. and whose area is897ift. 7. How many planks 12 ft. 6 in. long, and 8| in. wide, will floor a roomSOft. by 16 ft.? 8. Find the area of a square building, whose side is 26 yds. 5 in. 9. An area, measuring 80ft, 6 in. by 8 ft. 9 in., is to be paved; what will it cost at the rate of 45'. 8^/. per sq. ft.? 10. Find the cost of a slab 5 ft. 7 in. long, and 8 ft. 8 in. broad, at Ss. per square foot. 11. Find the area of a floor which measures 18 ft. 6 in. by 12 ft. 3 in., and the expense of carpeting it at 3*. per square yard. 1 2. What will be the expense of painting the surfaces, which mea- sure respectively as follows? (1) 23 ft. 6 in. by 20 ft., at 4.9. 6d. per sq. yd. (2) 14 ft. 3 in. by 11 ft. 11 in., at Is. M. per sq. ft. (3) 13 ft. 6 in. by 8 ft. 9 in., at 7*. M. per sq. yd. 13. Work by Cross Multiplication each of the following examples, and prove the truth of each result by Vulgar Fractions. (1) 18 ft. 9 in. X by 14 ft. 7 in. / SQUArxi; AND CUBIC MEASURE. 171 (2) 23 ft. 8 in. x by 16 ft. 9 in, (3) 27 ft. G'. 9'' X by 5 ft. 3'. (4) 22 ft. li'l in. X by 10 ft. 7h ii. (5) 4 ft. G; 5''xby9ft. 4\7". (6) 75 ft. 7i in. X by 38 ft. 3^- in. (7) 5 yds. 2 ft. 2 in. 3 pts. x by .5 yds, 1 1 in. 7 pts. 14. How many yards of carpet J yd. wide will cover a room 40 ft. Gin. by 24ft. Gin. 15. What length of paper J of a yard wide will be required to cover a wall 15 ft. 8 in. long by 11 ft. 3 in. high ? IG. Find the cost of a carpet fyard wide at 3^. 9c?. a yard for a room 20 feet by 18. 17. Find the expense of carpeting the following rooms : (1) 12ft. 4 in. long, and 12ft. Gin, broad^ with carpet f yd. wide, at 4^. 6d. a yard. (2) 29^- ft. long, and 14^ ft. broad, v;ith carpet fryd. wide, at 3?. Gd. a yard. (3) 15 ft. 6 in. long, and 12 ft. 9 in. broad, with carpet 24 in. wide, at 7*. ^d. a yard. (4) 2G5ft. long, and 18 ft, broad, with carpet | yd, broad, at 3^. Ad. a 3^ard. (5) 19 ft. 7 in. long, and 18 ft. 11 in. broad, with carpet 25 in. broad, at 4^. M. a yard. 10. Find the content, and (when required) the cost, of the following: (1) A piece of timber, whose length, breadth, and thickness are respectively 54^ ft., 5 ft., and 2ft. 5 in., at 9c?. a solid foot. (2) A cube, whose edge is 1 ft. 8 in., at 6d. a solid inch. (8) Digging a cubical cellar, whose length is 12 ft., at dd. a solid yard. (4) A cistern 6 feet deep, having a square bottom of which each side is 2^ ft. (5) A wall 1000 ft. long, lOi ft. high, and 2 ft. U in. thick. (G) A cube, whose edge is 18 ft. 7' . 7''. 19. Find the number of feet and inches in the floor, and the number of cubic feet and inches in the volume of a room 23 ft. 10 in. long, 18ft. 4 in. broad, and 11 ft. 3 in. high. 20. Find the length of paper, -|ths of a yard wide, required to cover the walls of a room, whose length is 27 ft. 5 in., breadth 14ft. 7 in., and height 12 ft. 10 in. 1 72 ARITHMETIC. 21. What would be the cost of painting the four walls of a room whose length is 24 ft. 3 in., breadth 15 ft. 8 in., and height 11 ft. 6 in., at 4*. a square foot ? 22. Find the expense of painting the walls and ceilings of each of the tirst two, and the walls of each of the last two of the following rooms : (1) A room whose length is IGft. 8 in., breadth 15 ft. 9 in., and height 14 ft., at I*, a sq. yd. (2) One whose length is 15 ft., breadth 10 ft., and height 9 ft. 9 in., at 1^. 4d. a sq. yd. (8) One whose circuit is 4ih ft., and height 8 ft. 5 in., at lie?, a sq. yd. (4) One whose circuit is 72 ft., and height 10^ feet, at lOM. a sq. yd. And find also the expense of papering the walls of the first two of the above rooms with paper 1 ft. 9 in. wide, at the following prices — the first at Ss, 6d. a yard, and the second at 1^. 2d a yard. 23. The length, breadth, and height of a room are 7 yds. 1ft. 3 in., 5 yds. 2 ft. 9 in., and 4 yds. 6 in., respectively. What length of paper two feet broad will be required to cover the walls, and what will it cost at M. per yard ? 24. Supposing the cost of a carpet in a room 25 feet long, at 55. a square yard, to be £6. 5s., determine the breadth of the room. 25. In a rectangular court, which measures 9G ft. by 84 ft., there arc four rectangular grass-plots, measuring each 22.} ft. by 18 ft. ; find the cost of paving the remaining part of the court at S^d. per square yard. 2G. If a piece of cloth be 94} yds. long, and l^^yds. broad, how broad is a piece of the same content, whose length is 741 yds. ? 27. How many sq. ft. and sq. in. remain out of 313 sq. ft. of carpet- ing, after covering a room 16 ft. 9 in. by 12 ft. 11 in. ? What is the price of the requisite carpeting at 3*. 6c?. a yard ? 28. On laying down a bowling-green with sods 2 ft. 6 in. long by 9 in. wide, it is found that it requires 75 sods to form one strip extend- ing the whole length of the green, and that a man can Jay down one strip and a quarter each day : find the space laid down in 8 days. 29. A piece of land, whose length is 151 yds. lift., and breadth SQUARE AND CUBIC MEASURE. 173 35 yds., is to be exchanged for part of a strip of land of the same quality, whose breadth is 15 yds. 21 ft. Find the length of the equivalent strip. SO. Find the difference between the content of a floor 80 ft. 9 in. long and 65 ft. 6 in. broad, and the sum of the contents of three others, the dimensions of each of which are exactly one-third of those of the other, 81. A reservoir is 24ft. 8 in. long, by 12ft. 9 in. wide ; how many cubic feet of water must be drawn off to make the surface sink 1 foot? 32. Divide 1532 ft. 9,% in. by 81 ft. 9 in. : and find the breadth of a room, the length of which is 17i- ft., and the area 250f ft. 33. How many sq. ft. of glazing are contained in the windows of a house of 4 stories, each story containing 12 windows, the breadth of each window being 3 ft. 6 in. ; the height of the windows on the ground and first floors being 71- ft., on the second floor 6 ft. 10 in., and on the thh-d floor 6 ft. 1 What will the cost be at lOcf. a sq.ft.? 34. How many bricks will be required to build a wall 20 yds. long 7^ ft. high, and 14 in. deep ; supposing a brick to be 9 in. long, 3J- in. broad, and 2;^ in. deep ? 35. How many tons of water are there in a cistern 18 ft. 8 in. long, 18 ft. 4 in. broad, and 6 ft. 9 in. deep, supposing a cubic foot of water to weigh 1000 oz.. ? S6. How many rods of brickwork are there in a wall 77 ft. long, 16 ft. high, and 1 ft. lOj in. thick 1 37. Find the expense of painting the outside of a cubical iron chest, whose edge is 2 ft. 5 in., at 1*. od. per sq. yd, 38. What will the painting of a room cost which is 20^ ft. long, I8Ht. broad, and 10 ft. high, containing 2 windows whose dimen- sions are 7 ft. by 4 ft. each, at the rate of 2*. 6d. per sq. yd. ? 39. A piece of cloth 5 times as long as broad cost i;i9 ; supposing the price of cloth to be 4*. 9d. a square yard, find the dimensions of the piece. 40. What length must be cut off a straight plank lift, broad, and f ft. deep, in order that it may contain 111 cubic ft. ? 41. A Turkey carpet, measuring lift. 6 in. by 9 ft. 8 in., is laid down on the floor of a room measuring 14 ft. by 12 ft. 6 in. ; deter- mine the quantity of Brussels carpet^ f yd. wide, which will be 174 ARITHMETIC. required to complete the covering of the area ; what will he the cost of it at Ss. 9c/. a yard ? 42. Shew hy Cross Multiplication and hy Vulgar Fractions how many cubic feet are contained in a beam 20 ft. 4 in. long, 1 ft. 5 in. broad, and 10 in. thick. 43. If 69 yds. of carjjet, f yd. wide, cover a room which is lO^yds. long, find the width of the room. 44. If a postage stamp be an inch long and |ths of an inch broad, how many stamps will be required for papering a room 16 ft. 10 in. long, 15 ft. 9 in. broad, and 12 ft. 6 in. high ? 4o. The length, width, and height of a room are respectively o6 ft., 24 ft., and 20 ft. ; how many yards of painting are tliere iii the walls of it, deducting for a fire-place 6 ft. by 5i-ft., and two win- dows, each 7| ft. by 3J ft. ? What would it cost to paper the above room with paper 2-^ft. wide, at lie?, a yard ? 46. How many bricks, each 9 in. long, 41 in. wide, and Sin. thick, will be required for a wall 100 yds. long, 15 ft. high, and 1ft. 10|^ in. thick ? ^ 47. A gentleman has a garden 200 ft. long and 180 ft. broad, and a gravel walk is to be made to run lengthways across it ; how wide must the path be so as to take up ^th of the garden ? 48. A wall is to be built 15 yds. long, 7 ft. high, and 18 in. thick, con- taining a doorway 6 ft. high, and 4 ft. wide. How many bricks will it require, the solid content of a brick being 108 cubic inches? 49. What v^'ould be the cost of paving a road of a uniform breadth of 4 yards extending round a rectangular piece of ground, the length of which is 85 yds., and breadth 56 yds., the cost of paving a square yard being 1^. 2d. ? do. How many paving-stones, each of them a foot long and 3^ of a foot wide, will be required for paving a street 45 ft. wide, sur- rounding a square, the side of which is 225 ft. ? 51. What will be the expense of paving a rectangular court-yard, whose length is 126 ft. and breadth 98 ft., with pebbles, at 9d. per sq. yd. ; and by how much will the expense be increased if a granite path, 5^ ft. wide, at lOs. 6d. per sq. yd., be laid down all round between the outside walls and the pebbles ? 62. A gentleman wishes to raise his lawn (vvhich is 1902ft. long and 1020 ft. broad) 2ft., and for that purpose digs a moat round it SQUARE AND CUBIC MEASURE. 175 17 yJs. broad in every part ; supposing the depth of the moat to be uniform, how deep must it be in order that he may have soil sufficient for his purpose ? 63. Find the expense of lining a cistern, 10 ft. 3 in. long, 6 ft. G in. broad, and 5 ft. 4} in. deep, v/ith lead, at £2. 2s. a cwt., which weighs 8 lbs. per sq. ft. 64. How many imperial gallons will a cistern contain whose length, depth, and breadth are 7 ft. 3 in., 3 ft. Sin., and 2 ft. 10 in. respec- tively ? 141. Examples wliich are usually classed under particular Rules, such as the Rule of Three, &c., can nevertheless be readily solved in- dependently by means of the foregoing principles. The following examples, which are worked out, are intended to exem- plify various methods of reasoning. In the examples for practice which follow them, questions will he found the solution of which may be easily arrived at in a similar way : the number of such questions in this place must necessarily be very limited, and therefore the student is strongly recommended to apply to all questions which are hereafter classed under particular Rules, an independent method of solution, as well as the one denoted by the Rule to which they are respectively affixed. Ex. 1. Express a degree (69|- m.) in metres, 82 metres being -^ 85 yds. 85 yards = 82 metres, 82 /. 1 yard = — metres ; .♦. 1 degree = (69^ x 1760) yards = (189 x 880) yards, /l;39x880x82\ ^ nnoo^'^ . = I i^- j metres = 11183of metres. Ex. 2. If §rds of a lottery ticket be worth £'220, what is the value of ■^ths of the same ? .*. §rds of the ticket ==X220. .'. ^-rd of the ticket = £110. .-. whole ticket = £(110 x 8) = £880. .-. fi-ths of the ticket =:xi- of £380 - £ — J^ ^ £90. Ex. 3. A person has -^ths of an estate of 4000 acres left him ; he sells |rds of his share : how many acres has he remaining, and what fraction of the whole estate will they be ? 176 ARITHMETIC. 2 3 2 He sells | of =: of 4000 acres, or f- of 4000 acres, (3 2 \ =- of 4000 -s: of 4000 ) acres = = of 4000 acres = 57lf acres. Ex. 4. The sum of £463. IGs. is to be raised in a parish, the assess- ment of which is £6184 ; what is the rate in the £.? £6184 produce £463f or £ ^^, J ^ /2319 1 \ /2319 1 „ \ • • ^^- P^^^"^^^ ^ K~^ " 6r84> '' KT " 6184 " 2^} *• 2319 , ^ , or ^-TTTv*., or 1^. 6rf. 1546 Ex. 5. After taking from my purse \ of my money, I find that § of what is then left amounts to ^s, 6cf. ; what money had 1 in my purse at first? Let unity, or 1, denote the sum in the purse at first. After takmg J^way i, f remains. Now by the question 2 3 2 3 ;, of - of unity, or - of - of the sum in the purse at first = 7«. 6i. or — of the sum in the purse at first = 7*. 6af., .*. sum in the purse at first = 15*. Ex. G. A met two beggars, B and C; and having -^ ^'^~hT ^^^ 1 o g of a moidore in his pocket, gave 5 = of - of that sum, and G -=oi the remainder ; what did each receive ? 40 75 ^hadatfirst|of^of^^of27...orL^. 7 2 B received = of ^ of ;r- *., or ^ *., or Gc/. A had left afterwards ( -;r ~ o ) *• = fi" *•» ,', C received -z oi~ s., or s" *•» or 2«. 6i. o b .. ^ MISCELLANEOUS EXAMPLES WORKED OUT. 177 Ex. 7. A farmer pays a corn-rent of 5 quarters of wheat and 3 quar- ters of barley, "W^inchester measure : what is the money value of his rent, when wheat is at 60.9., and barley at 54*. per quarter, imperial measure ; o2 imperial gallons being =33 Winchester gallons? Rent is 5 qrs. of wheat Win. mea. + 3 qrs. of barley Win. mea. 32 But 1 Win. gal.=gg imp. gal., 32 .', 1 "Win. < 4- ) guineas = 900 guineas. Ex. 17. Of a certain dynasty, |- of the kings are of the same name, J of another, ^ of a third, and yW of a fourth, and there are 5 besides : how many are there of each name ? Representing the whole dynasty by unity^, or 1. -= number of kings of one name, o ■7 = of a second..., 4 ~= of a third-. •, o ^ = of a fourth... .*. whole dynasty — ^, or 1 — — , or — = no. of remaining kings in it. But by the question, ^ of unity, or — of the whole dynasty —5 ; 24 .•. 1, or the whole dynasty, =5 x — -24; o .'. there are 8 kings of the 1st name, 6of the 2nd, 3of the 3rd, and 2 of the4th. Ex. 18. A can do a piece of work in 5 days, B can do it in 6 days, and C can do it in 7 days; in what time will A, B, and C, all working at it, finish the work ? Find also in what time A and B working toge- ther, A and C together, and B and C together, could respectively finish it. Representing the work by unity, or 1. In one day A does - part of the work, o MISCELLANEOUS EXAMPLES WORKED OUT. 181 In one tlay B docs - imrt of the work, C does ^ ; ^ „ ^, /I 1 1\ 107 ,, 4+ B+Cdo(^- + - + ^), or ^7,pai-t; 210 .*. time in which A + B+ C would finish the work = Yof ^^y^ = yjj ^^ays= l}g^^ days. Again, in one day ^ + JB do (- + -j, or^r- , of the v/ork; therefore time in which they would finish it^^^T ^^ ^A ^^y^- 80 In like manner, it may he shewn that A and C would finish the work in 211 days ; and B and C in 3/^ days. Ex. 19. It being given that A and B can do a piece of work in 2^ days; that A and C can do the same in 2\l days ; and that B and C can do it in S^''^ days : find the time in which A, B, and C would do the w^ork : working, first, all together, secondly, separately. In one day A and ^ do ;— of the work, tjO 12 A and C do — , So 5 and C do ^ .'. by addition, ^ + 1^ + ;^) , or ^, of the work, 107" .'. in one day ^ =h i? + C do - 1 '^lO .-. time required = j^ = ^ days = l-i^ days. 210 Again, work done by ^ + ^+ C in one day-work done by ^+ C in one day, 111.- -, 107 13 1 or, work done by ^ m one day =— - — = - ; therefore time required, in v^'hich A would do the work, = 5 days. 182 ARITHMETIC. In like manner it may be shewn, that B would do the work in 6 days, and that C would do it in 7 days. Ex. 20. A cistern is fed by a spout which can fill it in 2 hours, how long would it take to fill it if the cistern has a leak which would empty it in 10 hours ? In one hour spout fills - of the cistern. .leak empties ~ Tlicrefore in one hour, when the spout and leak are both open, the part of the cistern filled by what runs in — what runs out, \2 10/ 5' 1 5 .•. time required for filling the cistern = -n hrs.= 2hrs. = 2|^ hrs. 5 Ex. 21. A can perform a certain quantity of work in 5 days, B twice as much in 6 days, and C 4 times as much in 9 days ; in what time can A, B, and C working together, perform a piece of work 11 times as great ? In one day A does - of the work, B does 7^ or - 6 3 C does -^ ; .-. in one day .4 + 5 + C do ("k + o + q) ®^ 45 ^^ ^^^^ work, 45 ;. they would finish this piece of work in — days, (45 \ — X 11 J or 1\\ days. Ex. 22. A and B can do a piece of work in 15 and 18 days respect- ively ; they work together at it for 3 days, when B leaves, but A continues, and after 3 days is joined by C, and they finish it together in 4 days ; in what time would C do the piece of work by himself? MISCELLANEOUS QUESTIONS AND EXAMPLES. 183 Representing the work by unity, or L In one day A + B do (— + y^) of the work, in 3 days they do T— + ^j x3 n or 19 ^^30- . . „^ of the Avoik remains to be done. 3 1 In 3 days more A does rz or - of the work ; ••, when A is joined by C, =-— r , or j;- of the work remains to be done. oO 5 60 4 In 4, days more A does ~i of the work ; /. work which has to be done by C in 4 days _13_^ _ ^-1. ~ 80 15 ~ SO ~ ' .*. part of work to be done by C in one day = ^ , .*. time in wliich C would do the whole work = 24 days. E5C. LV. Mucellaneous Questions and Examples on preceding Arts. I. 1. State the rules for the multiplication and division of decimals, and divide 34*17 by 31. 2. A^hat is the value in English money of 155685 francs, when the exchange is at 24'25 francs per £ % 3. Reduce ^ + ^+^ + ^ to a decimal fraction. What decimal of a cwt. is 1 qr. 7 lbs. ? 4. Explain the principle of the Rule of Practice. Find (by Practice) the cost of 365^ tons of coals at 13*. 6\d. a ton; and the rent of 316 ac, 3 ro., 7 po. at £1. IGs. M. an acre. 5. If f of an estate be worth £1003. 17^. Id., what is the value of f of it? 184 ARITHMETIC. G. If a bankrupt pay S*. 4c?. in tlie pound, what will be received on a debt of £3678. 16*. ? 7. A person possessing ^ of an estate, sold ^ of ^r:; of his share for £120f ; what would |- of yg- of the estate sell for at the same rate ? 8. A man, his wife, and 8 children earn £1. 7s. Gd, a week ; the wife earns twice as much as each child, and the man three times as much as his wife ; required the man's weekly earnings. 9. If £1. sterling be worth 12 florins, and also worth 25 francs, 5Q centimes ; how many francs and centimes is one florin worth ? (lOO centimes = 1 franc.) 10. The wages of 5 men for 6 v/ceks being £14. 5^., how many weeks will 4 men work for £19? 11. 1. What is meant by saying that one sum is a certain fraction (for example f ) of another ? If 26 francs are equivalent to a pound, what fraction of a shilling is a franc ? Give the reasons for the process which you adopt in answering the question. 2. Express | of If of a mile in terms of a metre, supposing 32 metres = 35 yards. 8. A, B^ and C rent a pasture for £40. A puts in 8 cattle, B, 9, and C, 11 : how much should each pay for his share ? 4. Reduce 3fd to the decimal of 10^., and divide the result by 12"5. Explain the process employed. o. Find the value of 45 ac, 8ro., 20 po. at £111. 11^. 4d. per acre, by Practice. 6. If the property in a town be assessed at £60000, what' must be the rate in the £ in order that £2500 may be raised 1 7. If the circumference of a circle = Diameterx 314159 J find the number of revolutions passed over by a carriage.- wheel 5 ft. in diameter in 10 miles. 8. A farmer has to pay yearly to his landlord the price of 7^ bushels of wheat -at 4s. 9(/. per bushel, and 9^ of malt at 5^. 8-hat has pre- ceded, that by the expression 8 : 4 ;: 9 : 12, it is meant in fact that 4 12' 150. In order to form a proportion four numbers are required. It maj indeed happen that the second and third are the same, in which particu- lar case it might be said that only three numbers are required ; thus 9 : 6 : : 6 : 4 ; but er 3n in such a case it is better to consider the second and third as distinct numbers, and to regard the proportion as consisting of four numbers, of which indeed two are equal. The four numbers required to form a proportion are called its terms. In the proportion 3 : 4 : : 9 : 1 2, we have 3 for the first term, 4 for the second , 9 for the third, and 12 for the fourth term, of the proportion. 151. It has been stated that proportion is the equality of two ratios, and we have explained that the two numbers constituting a ratio must either be both abstract, or (if concrete) both of the same kind. In a proportion if one of the ratios be formed by two abstract numbers, the other may arise from two concrete numbers. For it has been explained (Art. 147) that if a ratio consist of two concrete numbers, we may reduce them both to the same denomination, and then treat the resulting numbers as abstract, the ratio of those abstract numbers being the same as that of the two concrete numbers from which they have arisen, For the same RULE OF THREE. 195 reason, one of the two ratios constituting a proportion may be formed from concrete numbers of one kind, while tlie other is formed from concrete numbers of a different kind ; for 7 days : 13 days :: 7 miles : 13 miles, each ratio being in fact that of 7 to 13. Indeed it appears by (Art. 147) that the ratio of two concrete numbers may always be ex- pressed by a ratio of two abstract numbers. If both or either of the ratios in a proportion be formed from concrete numbers, we may thus replace each such ratio by one arising from abstract numbers, and in this way every term of the proportion will become an abstract number; so that, notwithstanding the remark in note (Art. 2G), any one of the terms may then be multiplied or divided by any other. 152. It is readily seen that if proportion exist among four numbers taken in a certain order, it will exist also among the same numbers taken in the contrary order. Thus the numbers 8, 9, 24, 27, being pro- portionals in the order in vrhich they stand, the numbers 27, 24, 9, 8, will also be proportionals. For, 8 24. 9 ~ 27 ' .'. 1- . 8 • 0~ 1^ 24 '27' orl 9 :lX 27 24' 9 or- = 27 ^24 ; .*. 27 24 9 '8 ■ or 27 :24 :: 9 : 8. 27 It is apparent also from (Art, GQ) that ^7 = k • 153. If only three of the numbers in a proportion be given, we can by means of them tind the fourth, and the method or Rule by which it may be found is one of great importance in Arithmetic. We have seen that proportion exists among the numbers 8, 9, 24, 27. If the first three numbers only were given, and we were required, by means of these, to find the fourth, the method or Rule to be adopted ought to determine a number to which 24 would have the same ratio, as 8 to 9; or, which is seen from the last article to be the same thing, it ought to determine a number which will have the same ratio to 24, which 9 has to 8; this number being of course 27. Almost all questions which arise in the 13—2 196 AEITHMETia common concerns of life, so far as they require calculation by numbers, might be brought within the scope of the Rule of Three, which enables us to find the fourth term in a proportion, and which, on account of its great use and extensive application, is often called the Golden Rule, 154. I'hc Rule of Three, tlien, is a method by which we are enabled, from three numbers which are given, to find a fourth which shall bear the same ratio to the third as the second to the first, that is, shall be the same multiple, part, or parts of the third, as the second is of the first ; in other words, it is a Rule by which, when three terms of a pro- portion are given, we can determine the fourth. As most of the practical cases in which this Rule is made use of relate to concrete numbers, we shall express the Rule with especial reference to such cases, adding hov/ever a short direction for cases in which abstract numbers only are concerned. 155. Rule. "Leaving out of consideration superfluous quantities, find, out of the three quantities which are given, that which is of the same kind as the fourth or required quantity ; or that which is dis- tinguished from the other terms by the nature of the question : place this quantity as the third term of the proportion. " Now consider whetlier, from the nature of the question, the fourth term will be greater or less than the third ; if it be greater, then put the larger of the other two quantities in the second term, and the smaller in the first term ; but if less, put the smaller in the second term, and tho larger in the first term. " Take care to reduce the first and second terms to one and the same denomination, and also to reduce the third so that it may be wholly in one denomination ; remembering, however, that if the quantities involved be all of the same kind, it is unnecessary to reduce all the three terms to the same denomination, but only the first and second terms to one and the same denomination, and the third to a single denomination, which will not necessarily be the same as the former. When the terms have been properly reduced, multiply the second and third together, and divide by the first, treating all three as abstract numbers. The quotient will be the answer to the question, in the denomination to which the third term was reduced." If the case be one in which abstract numbers only are concerned, the question itself will show at once which of the numbers will form the third term of the proportion : the second and first will be determined aa RULE OF THREE. 197 above explained ; and then the answer to the question will be found by such multiplication and division as are directed in the Rule. The anangement of the given terms in the manner mentioned at the beginning of the llule, is commonly called stating the question. Sometimes a word or two, or a letter, or other symbol, will be added to represent the fourth or required term. Note 1. The process denoted by the above Rule may often be much abbreviated by dividing tlie first and second, or the first and third terms, (but never the second and third) by any number which will divide each of them without a remainder, and using the quotients instead of the numbers themselves. For, 9 : 12 :: 21 : 28 is the same as ^ = ||> which is the same as 1 = 1^, Avhich is the same as 3 : 4 :: 21 : 28, which represents the first proportion after its first and second terms have each been divided by the same number 3. Again, : 12 :: 21 : 28 is the same as y-^^iij which is the same as y%^ = v/y, which is the same as 3 : 12 :: 7 : 28, which represents the first proportion after the first and third terms have each been divided by 3. Again, 9 : 12 :: 21 : 28 is the same as ^=|l, but this is not the same as f = /yj which is the same us 9 : 4 :: 7 : 28, which represents the first proportion after the second and third terms have each been divided by 3. Moreover ^ is not equal to o-%^ and of course 9 : 4 :: 7 : 28 is not a true proportion. Note 2. Although w^e have said In the Rule, multiply the second and third terms together and then divide their product by the first ; it will be found in most cases advisable not to perform the actual multipli- cation until we have discovered, by putting the expression in the form of a fraction, whether there be any factor or factors common to the numerator and denominator, and if so, have rejected such factor or factors. 15G. It may be proper to observe that the Rule of Three is applicable in two different kinds of cases, according to which it is called the Rule of Three Direct or the Rule of Three Inverse. The method just stated (Art. 155) is applicable to both kinds of cases ; but as the distinction between the two is commonly noticed by writers on Arithmetic, it will be right to show in what it consists. The Rule of Three Direct is that in which more requires more, o? less requires less; or, in other words, in which a greater number requires a greater answer^ or a loss number a less answer. Thus in the question. 198 ARITHMETIC. " If 4 acres of land cost £250, find the cost of 15 acres, after the same rate." The 1 5 acres being more than the four acres, will requh-e a larger sum than £250 for their purchase, and so, in this case, more requires more. Again in the question, " If 15 acres of land cost £937. 10s. find the cost of 4 acres, after the same rate," the 4 acres being less than the 15 acres, -will require a less sum than £937. lO.s. for their purchase, and therefore, in this case^ less requires less. Such cases belong to the Rule of Three Direct. The Rule of Three Inverse is that in which more requires less, or less requires more : or, in other words, in which a greater number requires a less answer, or a less number a greater answer. Thus in the question, " If 4 men can mow a certain meadow in 3 days, find the time in which 6 men ought to mow it," the six men being more than the four, should perform the work in less time, and so, in this case, more requires less. Again, in the question, " If G men can mow a certain meadow in 2 days, find the time in which 4 men ought to mow it," the 4 men, being fewer than the 6, will require a longer time for performing the work, and therefore, in this case, less requires more. Such cases belong to the Rule of Three Inverse. llule of Three Direct. Ex. 1. Find the value of 87 yards of silk, when 25 yards cost £4. 7*. Qd. There are here three given quantities, 25 yards, 87 yards, and £4. 7*. Qd., and we have to find a fourth which will be the price of 87 yards. It is manifest that the three given quantities, 25 yards, 37 yards, £4. 7*. 6c?., and the required sum, must form a proportion, because the 25 yards must have the same relation in respect of magnitude to the 37 yards, which the £4. 7*. Qd. (cost of 25 yards) has to the required sum (cost of 87 yards). Proceeding then by Rule (Art. 155) we observe that the £4. 7^. Qd. is of the same kind as the required term, viz. money ; Ave make that the third term of the proportion ; and since the required sum (cost of 87 yards) must necessarily be greater than £4. 7*- Qd. (cost of 25 yards), w^e make 37 the second term, and 25 the first. We have thus the first three terms arranged as follows : 25 yds. : 87 yds. :: £4.7*. Qd. And the entire proportion will be as follows : 25 yds. : 87 yds. :: £4. 7*. Qd. : required cost. RULE OF THREE, 190 Tlie first and second terms are in one and the same denomination, and require no reduction. The third and fourth must be reduced to the lowest denomination in either of them, namely pence. Tlien since £4. 7*. Gd. — 1050 pence, the proportion becomes 25 yds. : 87 yds. :: 1050 pence : no. of pence in required sum. yVnd by our rule we must now treat the numbers as abstract, multiply the second and tliird together, and divide by the first. 1050 __87 7350 3150 88850 7770 1554 The quotient 1554 gives the number of pence in the required sum ot money, that being the denomination to which the third term was re- duced. We must now then reduce the 1554 pence to pounds, shillings and pence. 1554r/. ■Gd. 12 2,0 12,9- £6. 9s. Gd. therefore the required answer is £G. 9s. Gd. The above process would in common use be more compendiously wi'itten down as follows ; to-:i-j £. s. d. :: 4 . 7 . G 20 87 12 1050 87 7350 3150 (5 38850 7770 12 1554d 20 129^. ed. £G. 9s. Gd. 200 ARITHMETIC . Reason for the above process. We have the cost of 25 yards given, viz. £4. 7^. Qd., in ordet to enable us to find the cost of 87 yards. It is manifest that the required sum must have the same relation iu respect of magnitude to £4. 7^. Qd., which 37 yards have to 25 yards ; that is, the ratio of the required sum to £4. 7^^. Qd., or of the number of pence in the required sum to 1050 pence, must be equal to that of 87 yards to 2o yards. Now the ratio of the number of pence in the required sum to 1050 pence, is the same as that of the abstract number which indicates how many pence the required sum contains to tlie abstract number 1050, and may (if the former number be called the required number) be expressed I ,, p ^. required number by the fraction —^ — j— — — . And the ratio of 87 yards to 25 yards is the same as that of the ab^ etract number 87 to i\\Q abstract number 25, and may therefore, in like manner, be expressed by the fraction — . required number _ 87 . **' 1050 25' , required number ^^^^ 87 ^^.^ required number x 1050 87 x 1050 or — = — , 1050 25 ' . , , 87x1050 .. . ^_. or requn-ed number =; — ^ — , (Art. 66), 1050 X 87 This result shows that if we arrange the three given terms, 25 yards, 87 yards, and £4. 7 note 3 ; and to effect the requisite multiplication or division or both, after the fraction has been so simplified. Ex. 4 If I can travel 198 miles by railway for £2. 95. Gd., hovv far at the same rate of charge ought I to be carried for £8. 0^. 10|c?.? £2. 9^. Qd. : £8. 0^. lOld. :: 198 m. : required distance. 20 20 49 IGO 12 594 12 1930 4 2376 4 7722 lequire -,. , 7722x198 ,, 8861x198 ., d distance = — ——- — miles = — -.■,„„ miles = ?|^- miles = 643fmiles- 643^. miles. Ex. 5. The annual poor's rates on a net rental of £365. 7*. 3re respectively 50 chains 40 links, and 56 chains 25 links. 44. In what time will 25 men do a piece of work Vv'hich 12 men ccn. do in 3 days ? 45. If '3 of 4'5 cwt. cost £11. ofl. oc, v/liat is the price per lb. '( 46. A piece of gold at £3. 17'>'. lO^cf. per oz. is worth £150 ; v4iat will be the worth of a piece of silver of equal weight at 54.?. Qd. per lb. ? 47. If a piece of building land 375 ft. 6 in. by 75 ft. 6 in. cost £118, 2.9. C)}d., M'hat will be the price of a piece of similar land 278 ft. 9 in. hy 151 feet ? 48. A servant enters on a situation at 12 o'clock at noon on Jan. 1, 1854, at a yearly salary of So guineas, he leaves it at noon on the 27th of iMay following ; what ought he to receive for his services ? 49. A was owner of -^j of a vessel, and sold -fj oi g- of his share for £i9.^ ; what was the value of if of f of the vessel ? 50. A exchanged with B 60 yards of silk worth 7*. od. a yard for 48 yards of velvet ; what was the price of the velvet a yard? 51. A person, after paying 7d. in the £ for income-tax on his in- come, has £1632. 18.s. lOd. remaining ; what had he at first? 52. If a person's estate be worth 3000 guineas a year, and the land- tax be assessed at 2s. d\d. in the £, what is his annual income ? 14 210 ARITHMETIO. 53. A watch is 10 minutes too fast at 12 o'clock (noon) on Monday, and it gains 8'. 10'' a day ; what will he the time by th^ watch at a quarter joast 10 o'clock a.m. on the following Saturday ? 54. The circumference of a circle is to its diameter as 31416 ' 1 ; find (in feet and inches) the circumference of a circle whose diameter is 221- feet. 55. A bankrupt's estate amounts to £4:55. li. G^d., and his debts to £937. 10s. "What can he pay in the £ ? and what will a creditor lose on a debt of i:il4 ? 5(j. If the carriage of 8 cwt. cost 10*. for 40 miles^ how much ought to be carried for the same price for 2of- miles ? 57. n I spend 20 guineas in a fortnight, what must my income be that I may lay by £200 in the year 18oo ? 58. The house-tax upon a house rated at 175 guineas is £6. 17^. Q^dj what ^^ill be the tax upon one rated at £120 ? 59. A silver tankard, which weighs 1 lb., 10 oz., 10 dwts. cost £6. 3s. Qd.; what is the value of the silver per ounce ? 60. A man, working 71 hours a day, does a piece of work in 9 daya; how many hours a day must be work to finish it in 4^ days ? 61. If a pound of silver costs £8. Gs., what is tlie price of a salvet which weighs 7 lbs., 7 oz., 10 dwts., subject to a duty of 1^. 6d. per ounce, and an additional charge of Is. 10c/. per ounce for the workmanship ? 62. How much did a person spend in G3 days, who with an annual income of £818 is 90 guineas in debt at the end of a year 1 63. If 15 men, 12 women, and 9 boys, can complete a piece of work in 50 days, what time would 9 men, 15 women, and 18 boys take to do four times as much, the parts done by each in the same time being as the numbers 3, 2, and 1 ? 64. A person possesses £800 a year; how much may he spend per day in order to save £48. 2fi. 5c. after paying a tax of £^ on every £100 of income ? (}5. If 8 cows or 7 horses can eat the produce of a field in 29 days, ill how many days will 7 cows and 8 horses eat it up ? 66. How many yards of carpet f yard wide will cover a room whose width is 16 feet, and length 27A- feet ? 67. A person buys 100 eggs at the rate of 2 a penny, and 100 mor^ at the rate of 8 a penny : \yhat docs he gain or lose by selling them at the rate of 5 for 2d. ? RULE OF TliKEE. 211 G8, A church-clock is set at 12 o'clock on Saturday night ; at noon on Tuesday it is 3 minutes too fast : supposing its rate legular, what will be the true time when the clock strikes four on Thursday afternoon ? 69. A person after payuig a poors' rate of 10c?. in the pound has £728. Qs. M. remaining; what had he at first? 70. If a piece of work can be done in 50 days by 35 men working at it together, and if, after working together for 12 days, 16 of the men were to leave the work ; find the number of days in which the remaining men could finish the work. 71. A regiment of 1000 men are to have new coats ; each coat Is to contain 2^ yards of cloth 1} yards wide ; and it is to be lined with shal- loon of f- yard w^ide ; how many yards of shalloon will be required ? 72. If 5 ounces of silk can be spun into a thread two furlongs and a half long, what weight of silk would supply a thread sufficient to reach to the iMoon, a distance of 240,000 miles ? 73. How many revolutions will a carriage-wdieel, whose diameter is 8 feet, make in 4 miles ? (See Ex. 54.) 74. If 8 Qz. of sugar be worth •56255., what is the value of '7o of a ton? 75. The price of '0625 lbs. of tea is '45835.; what quantity can be bought for £61. 12^.? 76. Two watches, one of which gains as much as the other loses, viz. 2'. 5'' daily, are set right at 9 o'clock a. m. on Monday ; when will there be a difference of one hour in the times denoted by them ? 77. How many yards of matting, 2"5 feet broad, will cover a room 9 yards long, and 20 feet broad ? 78. A person bought 1008 gallons of spirits for £640 ; 48 gallons leaked out : at what rate must he sell the remainder per gallon so as not to lose by his bargain ? 79. If a soldier be allowed 12 lbs. of bread in 8 days, how mucli will serve a regiment of 850 men for the year 1853? 80. If 2000 men have provisions for 95 days, and if after 15 days 400 men go away; find how long the remaining provisions will serve the number left. 81. A gentleman has 10000 acres ; what is his yearly rental, it his weekly rental for 20 square poles be Ihd. ? (1 year ==52 weeks.) 82. If an ounce of gold bo worth £4-189583, what is the value of •8682291G lbs. ? 14—2 212 ARITHMETIC. 83. If 1000 men have provisions for 85 days, and if after 17 days 150 of the men go avray ; find how long the remaining provisions will serve the number left. 84. What is the quai'ter's rent of 182*8 acres of laud, at £4*65 per acre for a year ? 85. A grocer honght 2 tons, 8 cwt., 8 qrs. of goods for £120, and paid 50*. for expenses ; what must he sell the goods at per cwt. in order to clear £61. 5*. on the outlay ? 86. What must be the breadth of a piece of ground whose length is 40| yardsj in order that it may be twice as great as another piece of ground v/nose length is 14§- yards^ and whose breadth is ISfg- yards ? 87. If 3-75 yards of cloth cost £3-825, what will 88 yds,, 2 qrs., 8 nails cost? 88. Four horses and G cows together find sufficient grass on a certain field ; and 7 cows eat as much as 9 horses ; what must be the size of a field relatively to the former, which will support 18 horses and 9 cows? 89. A alone can reap a field in 5 days, and J? in 6 days, working 11 hours a day ; find in what time A and B can reap it together, working 10 hours a day. DOUBLE BULE OE THREE. 157. There are many questions, Avhich are of the same nature with those belonging to the Rule of Three, but which if worked out by means of that Rule as before given, would require two or more distinct applications of it. Every such question, in fact, may be considered to contain two or ]nore distinct questions belonging to the Rule of Three, and when each of those questions has been worked out by means of the Rule, the answer obtained for the last of them will be the answer to the original question. 158. The following example may serve to illustrate the preceding observations. *''If the carriage of 15 cwt. for 17 miles cost mc £4. 5s., what would the carriage of 21 cwt. for IG miles cost me V We observe that this question, though of a like nature with those which engaged our attention under the Rule of Three, is nevertheless of a more complicated description ; and the student, without further expla- nation, would find some difficulty in obtaining an answer to it by means of a single application of the Rule. For we observe, that instead of three given quantities, we have fi.ve, every one of which must necessarily DOUBLE RULE OF TlIREH. 213 liave a bearing on the answer, so that none of them can be snpci-fluous. If however the question be divitied into two distinct questions^ each of tliese, when superfluous terms are rejected, will be found to comprise only three given terms of a proportion, from which three terms the fourth is to be ascertained ; and the student would have no difficulty in working out each of these two questions by means of a single application of the Rule, so that in this way he will obtain the correct answer by ap- plying the Rule of Three twice over. The first question may be this ; '' If the carriage of 15 cwt. for 17 miles cost me £4. 5^., what would the carriage of 21 cwt. for 17 miles cost me V In this question the 17 miles would have no effect upon the answer, "because the distance is the same in both parts of the question, and the answer would clearly remain unaltered, if any other number of miles, or if the words "a certain distance," had been used instead of the 17 miles. This number may therefore be neglected as superfluous, and we have then three terms of a proportion remaining, and the fourth is to be found. Solving the question by the Rule of Three, we find that the answer will be £5. 19s. The second question may be this: " If the carriage of21 cwt. for 17 miles cost me £5. 19.?., what will the carriage of 21 cwt. for 16 miles cost me V In this question, for reasons similar to those before given, the 21 cwt. will be a superfluous quantity. Applying the Rule of Three to the question, we find the answer to be £5. 12.?. From the connection of the two questions with that originally pro- posed, we observe that £5. 12s.f thus obtained through two distinct applications of the Rule of Three, must be the answer to the original question. 159. We might give still more complicated instances, in which more than two distinct applications of the Rule of Three would be needed, in order to obtain the required answer j but the practical questions which most commonly occur, of the kind we have been treating of, would require only a double application of the Rule of Three, and, like the question which has been used by way of illustration, would comprise only five given quantities for the determination of a sixth which is not given. 160. The Double Rule op Three is a shorter or more compen- dious method of working out such questions as would require two or more applications of the Rule of Three; and it is sometimes called the Rule of Five, from the circumstance, that in the practical questiors to which it is applied, there are commonly five quantities given to find a sixth. 214 ARITHMETIC. 161. For the sake of convenience, we may divide each question into two parts, the supposition, and the demand: tlie former heing the part which expresses the conditions of the question, and the latter the part which mentions the thing demanded or sought. In the question, " If the car- riage of 15 cwt. for 17 miles cost me £4. 5^., what would the carriage of 21 cwt. for 16 miles cost me?" the words *'if the carriage of 15 cwt. for 17 miles cost me £4. 5.5.," form the supposition ; and the words, *' what would the carriage of 21 cwt. for 16 miles cost me V form the demand. Adopting this distinction we may give the following rule for working out examples in the Double Rule of Three, 161*. Rule. " Take from the supposition that quantity which corre- sponds to the quantity sought in the demand ; and write it down as a third term. Then take one of the other quantities in the supposition and the corresponding quantity in the demand, and consider them with reference to the third term only, (regarding each other quantity in the supposition and its corresponding quantity in the demand as being equal to each other); when the two quantities are so considered, if from the nature of the case, the fourth term would be greater than the third, then, as in the Rule of Three, put the larger of the two quantities in the second term, and the smaller in the first term ; but if less, put the smaller in the second term, and the larger in the first term. " Again, take another of the quantities given in the supposition, and the corresponding quantity in the demand ; and retaining the same third term, proceed in the same way to make one of those quantities a first term and the other a second term. " If there be other quantities in the supposition and demand, proceed in like manner with them. *' In each of these statings reduce the first and second terms to the same denomination. Let the common third term be also reduced to a single denomination if it be not already in that state. The terms maj' then be treated as abstract numbers. *' Multiply all the first terms together for a final first term, and all the second terms together for a final second term, and retain the former third term. In this final stating multiply the second and third terms together and divide the product by the first. The quotient will be the answer to the question in the denomination to which the third terai was reduced." Note. In dealing with the final statement obtained by our Rule, the two notes on Article 155 (see p. 197), '^vill often be found useful. DOUBLE RULE OF THREE. 215 Ex. 1. If a tradesman \vitli a capital of £2000 gain £50 in 3 months, how long will it take him with a capital of £3000 to gain £175 ? The 3 months in the supposition correspond with the quantity sought in the demand. ^Ve make the 3 months therefore the third term. Then taking the capital of £2000 in the supposition, and that of £3000 i.i the demand, and considering them Avith reference to the time in the tliird term, we see that if the amount of capital be increased, the time in which a given gain would be produced Avould be diminished, so that a fourth term would be less than the third ; therefore we place £3000 as a first term and £2000 as a second. Again, taking the gain of £50 from the supposition, and that of £175 from the demand, and considering them in like manner with reference to the time in the tliird term, we see that ii: the amount of gain be increased, the time in which a given capital w^ould produce it, must be increased also, so that here the fourth term would be greater than the third ; and therefore w^e place the £50 as a first term, and the £175 as a second term ; thus Ave have the following statements : £3000 : £2000 £50 : £1/0 J Proceeding according to Our Rule, we have the follow^ing statenlcnt 3000x50 : 2000x175 :: 3, 2G0O5< 175x3 and the required number of months 8000 x 50 2x175 50 ^ 25 '' The required answer is therefore 7 months. Reason for the above process. The tradesman, with a capital of £2000 gains £hO in 3 months. Let us first find, ])y the Rule of Three, how long he would be in gaining £175 with the same capital. Thus £50 : £175 :: 3ra. : required time. Required time = ( — -- — j months. Since then the tradesman with a capital of £2000 would gain £175 /175 X 3\ in f — '— — j months let us next find, by the Rule of Three, hoAv long it 21G AllITHMETIO. would take him to gain the same sum with a capital of £3000, and we must have the answer to the oi'iginal question. Thus £3000 : £2000 :: ^-^4^ months : required time. (l75 X 3 \ -— — r — X 2000 1 -f- 8000 175 X 3 X 2000 ^^^ = ■ ^ -^8000 oO _ 1 7-5 X 3 X 2000 1 50 ^ 800 _ ]75x 3x2000 60 X 8000 2000 X 3 X 175 3000 X 50 whence it appears that if \ye arrange the quantities given by the question as follows : . £3000 J £2000 ) £50 : £175 ) " "' and treat the numbers as abstract ; and then multiply the two first terms together for a single first term, and the two second terms together for a single second term ; and then divide the product of the second and third terms by the first, we shall obtain the answer in that denomination to which the third term was reduced. Or thus : A capital of £2000 gains £50 in 3 months, £1 £50 in (3 X 2000) months, . /3 X 20or; £1 £1 in — £-— ' ) month --- -^"(1^)—. /2000 X 175 X 3\ V 8000x50 / ' that is, if we arrange the given quantities as follows, £3000 : £2000 £50 : £175 3 m, DOUBLE RULE OF THREE. 217 vre ohtain the required time in months by multiplying the two first terms together for a final first term, the two second terms together for a final second term ; and then dividing the product of the second and third terms by the first term. Ex. 2. If a tradesman with a capital of £2000 gain £.50 in 3 months, what sum will he gain ^Yith a capital of £3000 in 7 months ? The £50 in tlie supposition corresponds to the quantity sought in the demand. Make this £50 the third term. Then taking the capital of £2000 in the supposition, and that of £3000 in the demand, and consider- ing them with reference to the gain in the third term, we observe that if the amount of capital be increased, so also will be the gain in a given time, and thus the fourth term would be greater than the third ; therefore we place the £2000 as the first term, and the £3000 as the second. Again, taking the 3 months in the supposition, and the 7 months in the demand, and considering them in like manner with reference to the gain in the third term, we observe, that as the time is increased, so also will be the gain from a given capital, and thus the fourth term would be greater than the third ; therefore Ave place the 3 months as a. first term, and the 7 months as a second. We thus obtain the following statements : £2000 : £3000 *• £50 m i 7m Proceeding according to our Rule, we obtain the following statement; 2000x3 : 3000x7 :: 50, , ,, . , . , 8000 X 7 X 50 and the required sum m pounds = — ^7— 3x7x50 2x8 The answer is therefore £175. --7x25 = 175. Ex. 3. If 7 horses be kept 20 days for £14, how many will be kept 7 days for £28 ? The 7 horses in the supposition correspond to the required quantity (number of horses) in the demand. Make this the third terra. Then, taking the 20 days in the supposition^ and the 7 days in the demand, and considering them with reference to our third term, we observe that if tire number of days be diminished, the number of horses which can be kept in them for a given sum of money wdll be increased, and thus a fourth term would be greater than the third ; we therefore place the 7 days in a first 218 ARITHMETIC. term, and the 20 days in a second. Again, taking the £14 in the supposition, and the £28 in the demand, and considering them with reference to the third term, we observe that if the sum be increased the number of horses which can be kept by it in a given time will be increased also ; so that here also a fourth term would be greater than the third ; we there- fore place the £14 in a first term, and the £28 in a second. We thus obtain the following statements : 7 days : 20 days } ^ , £14 : £28 1 '- ^ ^''''''' which, by our Rnle, will give the following single statement j 7x14 : 20x28 :: 7, 20 X 28 X 7 and thus, the required number of horses = — = — —. — / X 1^ = 40. The answer is therefore 40 horses. Ex, 4. If I get 8 oz. weight of bread for Gd. when wheat is 15s. a bushel, what ought a bushel of wheat to be when I get 12 oz. of bread for M. ? The price of a bushel of wdieat is required ; to this the 15^. in the supposition corresponds. Place this as the third term. Then taking the 8 oz. in the supposition and the 12 oz. in the demand, and considering them with reference to the price in the third term, we observe that tlie greater the w^eight of bread we obtain for a given sum the less will bo the price of a bushel of w^heat, and so a fourth term would be less than the third ; we therefore place the 12 oz. as a first term, and the 8 oz. as a second term. Again, taking the 6d. in the supposition and the Ad. in the demand, we consider that the less we pay for a given weight of bread, the less will be the price of a bushel of wdieat, so that here also a fourth term would be less than the third ; therefore we place the 6d. as a first term, and the Ad. as a second. Thus we have the following statements : 12 oz.: 8oz.| .. ^^^^ Cd. : Ad. J which, by our Rule, will give the following single statement ; 12x6 : 8x4 :: 15, and thus, the required price will be DOUBLE RULE OF THRFxE. 210 8x4x15 8x15 4x5 20 ^ _, ;: S. = — S. = S. = -- S.~ QS. on. 12x6 3x6 3 3 Ex. 5. If 20 men can perform a piece of work in 12 days, find the number of men who could perform another piece of work 8 times as great in ^th of the time. The first piece of work 'being reckoned as 1, tlie second must be reckoned as 3. The 20 men in the supposition must be taken as the third term. Then, taking the piece of work (represented by 1) in the supposition, and the piece of work (represented by 8) in the demand, we observe that if the work be increased the number of men to perform it in a given time must be increased, and we therefore place the 1 as a first term, and the 3 as a second. Again, taking the 12 days in the supposition and the 1^2 days in the demand, we observe that if the number of days be di- minished, the number of men required to perform any given work will be increased, and therefore we place the ^f days as a first term, and the 12 days as a secotid terra. Thus we have the following statements, '^ ' ^ I :: 20 men, ^ days : 12 days J which, by our Rule, will give the following single statement : V : 3x12 :: 20, md thus tlic required number of men will be 3 X 12 x20_ 3x12x20x5 ¥ "~ 12 300. Ex. 0. If 252 men can dig a trench 210 yards long, 3 wide, and 2 deep, in 5 days of 1 1 hours each ; in how many days of 9 hours each will 22 men dig a trench of 420 yds. long, 5 wide, and 3 deep ? The first trench contains (210 x 3 x 2) cubic yds. = 1260 cubic yds. The second (420 x 5 x 3) cubic yds. = 6300 cubic yds. On the supposition therefore that 252 men can remove 1260 cubic yds. of earth in 55 hours, we have to find in how many hours 22 men can re- move 6300 cubic yds. The 55 hours correspond to the quantity sought. Make this the third term. Then, taking the 252 men in the supposition, and the 22 220 ARITHMETIC. men in the demand, we observe that if the number of men be diminished, the number of working hours in which a given work can be performed will be increased, and we therefore place the 22 men as a first term, and the 252 men as a second. Again, taking the 1260 cubic yds. in the sup- position and the G300 cub. yds. in the demand, we consider that if the number of cubic yds. be increased, the number of working hours in which a given number of men can perform the work will be increased also, and therefore we place the 1260 cubic yds. as a first term, and the 6300 cubic yds. as a second. Then we have the following statements : 22 men : 252 men | 1260 cub. yds, : G800 cub. yds. j '• ^^ ^^^^^^' which, by our Rule, will give the following single statement : 22x1260 : 252x6300 :: 55, and thus the required time 252 X 6300 X 55 , , , ==-2271-260-™'^^"'^^^'^'' _ 252 X 5 X 55 " 22 ?= 3150 working hours = — ^ days of 9 working hours ^ 350 such days. Ex. 7. If 4 men earn £15 in 20 days, how many men will earn 10 guineas in 7 days ? £15 : 10 guineas ) w 1 OA J / :: 4 men. 7 days : 20 days J The £15 and the 10 guineas, being in different denominations, must, in accordance with our Rule, be reduced to one and the same denomina- tion. Thus, £15 being =3005., and 10 guineas being =210^., we have 3006\ : 2105. ) ^' J OA J V :: 4 men, 7 days : 20 days ) ' which, by our Rule, gives the following single statement : 800x7 : 210x20 :: 4, work ins: hours DOUBLE IIULE OF THREE. 221 210 X '^O X 4 and thus the required number of men = — _ 21 X 2 X 4 ~ 8x7 =^2x4 Tlie answer therefore is 8 men. Ex. 8. If 5G0 flag-stones, each 1^ feet square, will pave a court-yard, how many will be required for a yard twice the size, each flag-stone being 14 In. by 9 in. ? Superficial content of each of former flag-stones - (U X U) sq. ft. = (§ X f ) sq. ft - a gq. ft. Superficial contsnt of each of the latter flag-stones = (M X fV) sq. ft. = a X f ) sq- ft. - 1 sq. ft. Considering the first court-yard as 1, and therefore the second as 2, out statements will be |- sq. ft. : f sq. ft. ) "^ ^ J . 2 ('- o60 flag-stoues, which, by our Rulc;, will give us the follovWng' single statement : I : f x2 :: 560, and thus the required number of flag-stones -(fx2x560)-| ^(.^xSGOxf) .^^^^^2800. 2 X V Ex. 9. If 10 cannon, which fire 8 rounds in 5 minutes, kill 270 men in an hour and a half, hov/ many cannon, which fire o rounds in 6 minutes, will kill 500 men in one hour? The first 10 cannon, firing f of a round in a minute, kill 270 men in ^ hours. It is required to find how many camion, firing f of a round in a minute will kill 500 men in 1 hour. The 10 cannon in the supposition correspond to the quantity sought in the demand. We make this the third term. Then, taking the f of a round in the supposition and the f of a round in the demand, we ob- serve that if the part of a round which is fired in a minute be increased, the number of cannon for eff'ecting a certain slaughter would be dimi- nished ; and therefore we place the f of a round as a first term, and the 222 AmTHMETlC. f of a round as tlie second. Again^ taking the 270 men in the supposition and the 500 men in the demand, we observe that an increase in the number of men killed would require an increase in the number of cannon ; and therefore we place the 270 men as a first term, and the 500 men as a second. Again, taking the | hours in the supposition and the 1 hour in the demand, we consider that if the time in which a certain number of men are killed be diminished, the number of cannon would be increased; and therefore we place the 1 hour as a first term and the f hours as a second. Our statements will therefore be, |- round : f round ] 270 men : 500 men )■ :: 10 cannon, 1 hour : f hours ^ which, by our Rule, will give us the following single statement : I X 270x1 : I X 500x3- :: 10, or 5x45 i 8x50x3 :: 10, . , , » 8x50x8x10 ^^ **. required number of cannon = — — =20. ^ o x4o Ex. 10. A town which is defended by 1200 men, with provisions enough to sustain them 42 days, supposing each man to receive 18 oz. a day, obtains an increase of 200 men to its garrison ; v/hat must novv be the allowance to each man, in order that the provisions may serve the whole garrison for 54 days-? The 1400 men will belong to the demand : for tlie question is, what must be the allowance to each man, when the garrison is increased to 1400 men, in order that the provisions may last 54 days. The 18 oz. must clearly, according to our Rule, be the third term. Taking the 1200 men from the supposition, and the 1400 men from the demand, we consider that if the number of men be increased, the allow- ance to each must be diminished, in order that the provisions may last a given time ; and we therefore place the 1400 men as a first term, and the 1200 men as a second. Again, taking the 42 days in the supposition and the 54 days in the demand, we consider that if the number of days during which a garrison must be sustained be increased, the allowance to each man must be diminished ; and we therefore place the 54 days as a first term and the 42 days as a second term. Our statements wiil there- fore be, 1400 men : 1200 men ) 18 OS, 54 days : 42 days l^= DOUBLE RULE OF THREE. 223 which; by our Rule, 'svill give us the following single statement : 1400x54 : 1200x42 :: 18, . , ,, 1200x42x18 /. required allowance = 1400 x ^""" ^^• :-12oz. so that 12 oz. will be the answer. Ex. 11. If the carriage of 87 stone, Gibs, for 7 miles cost £2. 5s.. what weight should be carried 12 miles for £3. 106. ? 37 stone, 6 lbs. = 524 lbs. ; £2. 5s. - 45^. ; £3. 106\ - 70^. Our statements will be 12 miles : 7 miles } ^_, „ w^ ( '•'• 524 lbs., 45s. : 70s, ) which, by our Rule, give the following single statement : 12x45 : 7x70 :: 524, • 1 1 fn 7x70x524 .*. required number 01 lbs. = — — — — — ^ 12 X 4o ^475^ lbs. =t475lbs. 7oz. lloVdrs. ^83st. 18 lbs. 7oz.il/7- drs. so that the answer is 83 st. 18 lbs. 7oz. 11. ^V drs. Instead of reducing the quantities to low^er denominations, as in the above operation, we might have kept them in the higher denominations, by reducing any part which was expressed in a lower to a fraction of the higher denomination. Thus, observing that G lbs. — j^j st. = f st., and 5*. = £i, and 10s. = £^, we have 12 miles : 7 miles stone. 12x2^ ; 7x31^ :: 87f, 202 or, 12 xf : 7x^- ;: -^ , oT 49 2G2 . or, 27:^ ::-y; 49 262 49x202 ^ ,_^ „,^ /. since 2" ^ "t" = 9 ., j ~- = ^ X 131 = 91 / , 917 i-equlred weight = *-^ stone = 33 st. 13 lbs. 7 oz ll^ydrs. 224 ARITHMETIC. Ex. LVII. 1. If 7 men can reap 6 acres in 12 hourSj how many men will reap 15 acres in 14 hours ? 2. If 8 men earn £15 in 20 days, how many men will earn 15 guineas in 9 days, at the same rate ? o. If 16 horses eat 96 bushels of corn in 42 days, in how many days will 7 horses eat 6Q bushels ? 4. If 800 soldiers consume 5 sacks of flour in 6 days, how many v/ill consume 15 sacks in 2 days ? 5. If 17 bushels be consumed by 6 horses in 10 days, what quantity will 8 horses eat in 11 days, at the same rate ? G. 16 horses can plough 1280 acres in 8 days, how many acres will 12 horses plough in 5 days ? 7. If 11 cwt. can be carried 12 miles for £1. 5c. how far can 3G cwt. 23 lbs. be carried for £5. 2 fi. 5c. ? 8. If the carriage of 8 cwt. of goods for 124 miles be 6 guineas, what weight ought to be carried 53 miles for half the money ? 9. If 5 men on a tour of 11 months, spend £641. 13^. 4d., how much at the same rate would it cost a party of 7 men for 4 months ? 10. If with a capital of £1000 a tradesman gain £100 in 5 mont^.s, in what time will he gain £49. 5fl. with a capital of £225 ? 11. If it cost £59. 2s. l^c?. to keep 3 horses for 7 months, what will it cost to keep 2 horses for 11 months ? 12. The carriage of 4 cwt., Sqrs., for 160 miles costs £8. 8fl. 5c. j what weight ought to be carried 100 miles for £6. 0*. 8fd ? 13. If 1 man can reap 845# sq. yds. in an hour, how long will 7 such men take to reap 6 acres ? 14. If 20 men in 3 weeks earn £90,' in what time will 12 men eam £150? 15. If the carriage of 1 cwt., 8 qrs., 21 lbs. for 52j- miles come to 17'!>'. 5d., what will be charged for 2^ tons for 46^ miles ? 16. If 10 men can reap a field of 7^- acres in 3 days of 12 hours each, how long will it take 8 men to reap 9 acres, working 16 hours a day? 17. If 25 men can do a piece of work in 24 days, working 8 hours a day, how many hours a day would 30 men have to work in order to do the same piece of work in 16 days? DOUBLE RULE OF THREE. 225 18. If the rent of a farm of 17 ac, 3 ro., 2 po., be £29. 4.j. 7d., what would be the rent of another farm, containing- 2G ac., 2 ro., 23 po., if G acres of the former be worth 7 acres of the latter ? 19. If 1500 copies of a book of 12 sheets require G6 reams of paper, how much paper will be required for 5000 copies of a book of 25 sheets, of the same size as the former ? 20. If 5 men can reap a rectangular field whose length is 800 ft. and breadth 700 ft. in Sh days of 14 hours each ; in how many days of 12 hours each can 7 men reap a field whose length is 1800 ft. and breadth 960 ft. 1 21. If a thousand men besieged in a town with provisions for 5 weeks, allowing each man 16 oz. a day, be reinforced with 500 men more, and have their daily allowance reduced to 6§ oz. ; how long ^vill the pro- visions last them ? 22. If 20 masons build a wall 50 feet long, 2 feet thick, and 14 feet high, in 12 days of 7 hi's. each, in how many days of 10 hrs. each will 60 masons build a wall 500 feet long, 4 thick, and 16 high ? 23. If 10 men can perform a piece of work in 24 days, how many men will perform another piece of work 7 times as great, in one-fifth of the time ? 24. If 125 men can make an embankment 100 yards long, 20 feet wide, and 4 feet high, in 4 days, working 12 hours a day, how many men must be employed to make an embankment 1000 yards long, 16 feet wide, and 6 feet high, in 3 days, working 10 hours a day ? 25. "What is the weight of a block of stone 12 ft. 6 in. long, 6 ft. 6 in. broad, and 8 ft. 3 in. deep, when a block of the same stone 5 ft. long, 3 ft. 9 in. broad, and 2 ft. 6 in. deep, weighs 7500 lbs. ? 26. If 100 men drink £20 worth of wine at 4^. Qd. per bottle, how many men will drink £72. worth at 5s. per bottle, in the same time, at the same rate of drinking ? 27. If 5 horses require as much corn as 8 ponies, and 15 quarters last 12 ponies for 64 days, how long may 25 horses be kept for £41. 5s. when corn is 22 shillings a quarter ? 28. If 421 yds. of cloth which is 18 in. wide cost £59. 14^. 2d., what will 1181- yds. of yard- wide cloth of the same quality cost ? 29. 124 men dig a trench 110 yds. long, 3 ft. wide, and 4 ft. deep, in 5 days of 11 hours each; another trench is dug by half the number of 15 226 ARITHMETIC. men in 7 days of 9 hours each ; how many feet of water is it capable of holding ? 80. If the fourpenny loaf \veigh 8'35 lbs. when wheat is at 4'75s. a bus., what ought to be paid for 47Hbs. of bread when wheat is at 13*4,?. a bus. ? SI. A pit 24 ft. deep, 14 sq. ft. horizontal section cost £3 to dig out ; how deep will a pit be of horizontal section 7 ft. by 9 ft., which costs £4. 10s. ? 82. The value of the paper required for papering a room, supposing it I yard wide, and ^Id. a yard, is £2. 8s. Ihd. ; what would it come to, if it were 2 feet wide and M. a yard ? S3. 7 men working 16 days can mow a field of corn 1820 j^ards long and 880 wide ; what will be the length of the side of a field 1320 yards broad which 4 men can mow in 42 days ? 84. A beam 16 feet long, £1 feet broad, and 8 inches thick, weighs 1280 lbs. ; what must be the length of another beam of the same material, whose breadth is 8} feet, thickness 7^ inches, and weight 2028 lbs. ? 85. If 12 oxen and 85 sheep eat 12 tons, 12 cwt. of hay in 8 days, how much will it cost per month (of 28 days) to feed 9 oxen and 12 sheep, the price of hay being 4 guineas a ton, and 3 oxen being supposed to eat as much as 7 sheep 1 36. If 1 man and 2 women do a piece of work in 10 days, find in how long a time 2 men and 1 woman will do a piece of work 4 times as great, the rates of working of a man and woman being as 3 to 2. 37. A person is able to perform a journey of 142*2 miles in 4 J- days when the day is 10*164 hours long; how many days will he be in travelling 505*6 miles v.^hen the days are 8*4 hours long 1 38. If the sixpenny loaf weigh 4"85 lbs. when wheat is at 5*75*. per bushel, what weight of bread, when wheat is at 18*4^. per bushel, ought to be purchased for 18*135. ? 89. If a family of 9 people can live comfortably in England for 1560 guineas a year, what will it cost a family of 8 to live in Belgium in the same style for seven months, prices being supposed to be ^ of what they would be in Emjland ? SIMPLE INTEREST. 227 INTEREST. 162. Def. Interest is the sum of money paid for the loan or use of some other sum of money, lent for a certain time at a fixed rate; generally at so much for each £100 for one year. The money lent is called the Principal. The interest of £100 for a year is called the Rate per Cent. The principal + the interest is called the Amount. Interest is divided into Simple and Compound. When interest is reckoned only on the original principal, it is called Simple Interest. When the interest at the end of the first period, instead of being paid by the borrower, is retained by him and added on as principal to the former principal, interest being calculated on the new principal for the next period, and this interest again, instead of being paid, is retained and added on to the last principal for a new principal, and so on; it is called Compound Interest. SIMPLE INTEREST. 163. To find the Interest of a given sum of money at a given rate per cent, for a year. Rule. " Multiply the principal by the rate per cent., and divide the product by 100, as in (Art. 126)." Note 1. The interest for any given number of years will of course be found by multiplying the interest for one year by the number of years; and the interest for any parts of a year may be found from tha interest for one year, by Practice, or by the Rule of Three. Note 2. If the interest has to be calculated from one given day to another, as for instance from the 80th of January to the 7th of February, the 80th of January must be left out in the calculation, and the 7th of February must be taken into account, for the borrower will not have had the use of the money for one day till the 31st of January. N'ote 8. If the amount be required, the interest has first to be found for the given time, and the principal has then to be added to it. Ex. Find the simple interest of £250 for one year at 5 per cent, per annum. 15-2 228 ARITHMETIC. Proceeding according to tlie Rule given above, £. 250 5 £12-50 20 10-00^. therefore the interest is £12. lO-y. Reason for the Process. The sum of £100 must have the same relation in respect of magni- tude to £250 as the simple interest of £100 for a year has to the simple interest of £250 for a year; and thus the £100, £250, £5, and the required interest must form a proportion. (Art. 148.) "We have then £100 : £250 :: £5 : required interest, 250x5 ,. , _., (Art. 15o), 100 whencCj required interest = £ which agrees with the Rule given above. Examples worked out. Ex. 1. Find the simple interest and amount of £417. 7*. 9d for ] year, 10 months, at 4| per cent. 8. £. s. 417 . 7 d. 9 1669 . 11 156 . 10 . £18-26 . 1 20 5'2\s. 12 2-56^. '^ £. 417 . 7 d. . 9 3 1252 . 8 . 3 156 . 10 ^ .'. Int. for 1 year = 18 . 5 -18 . 5 Int. for 6 mo., or ,V of 1 year= 9 . 2 Int. for 4 mo., or ?. of 1 year =6.1 .-. Int for 1 yr., 10 mo -33 .9.6^ .-. amount = £417. 7^. 9^/. + £33. 9*. G^Ud. = £450. 17^. ^Ud. ,56| loo mi SIMPLE INTEREST. 229 Note. In examples like the above we may reckon 12 months to the year; but if Calendar months are given, the interest Avill then he best found by the Rule of Three ; as for instance in the following example : Ex. 2. Find the simple interest and the amount of £100. 13^. 4td. from June 15, 1843, to Sept. 18, 1843, at U per cent. £. 106 . 13 d. . 4 "^l 426 . 13 . 4 53 . 6 . 8 £4-80 . . . 20 16-00*. .*. £4. 16s. is the interest for 1 year. The number of days from June 15 to Sept. 13 ^15 + 31 + 81 + 18 ^95. Hence, 365 days : 95 days :: £4. 16*. : interest required, whence, it will be found, that interest required = £1. 4*. ll^d. ^q.; .'. amount = £106. 13*. 4d. + £l. 4*. U^d. f^q.=:=£107. 18*. S^d.^q. 164. Defs. Co3imission is the sum of money which a merchant charges for buying or selling goods for another. Brokerage is of the same nature as Commission, but has relation to money transactions, rather than dealings in goods or merchandise. Insurance is a contract, by which one party on being paid a certain sum or Premium by another party on property which is subject to risk, undertakes, in case of loss, to make good to the owner the value of that property. Questions on Commission, Brokerage, and Insurance, these charges l)eing usually made at so much per cent., amount to the same thing as finding the interest on a given amount at a given rate for one year, and may therefore be worked by the Rule given above for Simple Interest. There is, however, one case of Insurance which it may be well to notice by an example worked out. 230 ARITHMETIC. Ex. If goods worth £1200 be insured at £1. 10s. per cent., to what amount must they be insured, so tbat in case of loss the party insuring may recover the value of the goods and the premium ? If they be insured at their actual worth the premium paid will be lost, since the insurer will get £1200 only. But if every (£100 -£1. lOi.), or £98. 105., be insured for £100, then, in case of loss, the value of the goods £98. 10*. + £1. lO*. (the premium paid) will be recovered. Thus we have £981 : £1200 :: £100 : sum which is required to be insured ; whence, sum required to be insured =£1218. 5s. 6d. nearly. Ex. LVIII. 1. Find the simple Interest (1) On £85 for 1 year at 5 per cent, (2) On £310 for 1 year at 4 per cent. (3) On £1000 for 1 year at 4^ per cent. (4) On £475 for 3 years at 5 per cent. (5) On £986. Us. 3d. for 2 years at 4 per cent. (6) On £556. 13*. 4rf. for 6 years at 5 per cent. (7) On £945. 10*. for 2 years at 4 per cent. (8) On £198. 6s. 8d. for 1 year at 3i per cent. (9) On £236. Gs. 8d for 2} years at 3 per cent. (10) On £98. 15.S. 10^. for i year at 2| per cent. 2. Find the amount (1) Of £1000 for 2 years at 4} per cent. (2) Of £2833. es. M. for 4.^ years at 8 per cent. (3) Of £1050. 6fl. 2c. 5m. for 6 years at 41- per cent. (4) Of £139. 12?. Qd. for 31- years at 5^ per cent. (5) Of 1895 guineas for 4| years at 2| per cent. (6) Of £1534. Qs. Sd. for If years at 3} per cent. (7) Of £411, ]0s. for I year at 4^ per cent. (8) Of £1595. Ifl. 2 c. 5 m. for 5% years at 8| per cent. 3. Find the Simple Interest and Amount (1) Of £.375 for 3 years, 8 months, at 31- per cent. (2) Of £446. 10*. for 3 years, 3 months, at 5 per cent. (3) Of £220 for 7 months at 3J per cent. (4) Of £243. 10 >. for 2 years, 5 months, at 43- per cent. (5) Of 10 guineas for 117 days at 3^ per cent. SIMPLE INTEREST. 231 (6) Of £684. IQs. 8d. for 1 year, 11 months, at 4 J per cent. (7) Of 40 guineas from March 16, 1850, to Jan. 23, 1851, at 3| per cent. (8) Of £320. 15s. for 2 j^ears, 35 days, at 4| per cent. (9) Of £34. 10*. from August 10 to October 21, at 41- per cent. 4. Find the brokerage on £715. 12*. 6d. at 4^^ per cent. 5. What is the annual cost of insuring £4000 worth of property at h per cent. ? 6. What must be the sum insured at 4^ per cent, on goods worth £1910, so that in case of loss the worth of the goods and the premium may be recovered ? 7. At 7| per cent., what will be the cost of insuring property worth 500 guineas, so that in the event of loss the worth of the goods and the premium of insurance may be recovered ? 165. In all questions of Interest, if any three of the four {principal, rate per cent., time, amount) he given, the fourth may he found ; as, for instance, in the following examples. Ex. 1. Find the amount of £225 m 4 years at 2>h per cent, sim- ple interest. 225 31 675 112 . K) £7-87 . 10 20 17-50.9. 12 6-OOc?. .-. Int. for 1 year =-£7- 17*. Qd. 4 years = £31. 10*.; .-. amount is £225 + £31. 10*., or £256, 10*. Ex. 2. In what time will £225 amount to ^€256. 10*, at 8J per cent. simple interest ? £256. 10*. -£225. = £31. 10*., which is the interest to be obtained on £225 in order that it may amount to £256. 10*. But Int. of £225 for 1 year = £7. 17*. Qd. ; which must have the same relation in respect of magnitude to the £31. 10*. as the 1 year has to the required time ; 232 ARITHMETIC. .'. £7. l7s. Gd. : £31. 10^. :: 1 year : required number of years, whence, required number of years = 4. Ex. 3. At what rate per cent., simple interest, will £225 amount to £266. 105. in 4 years ? In other words, at what rate per cent, will £225 give £31.105, for ... £.31.105. ^^ ,^ ^; . « mterest m 4 years, or — , or £/. 1^5. Gd. m one year i Then £225 : £100 :: £7. 17^. 6d. : required rate per cent., whence, required rate per cent. = 3^^. Ex. 4. What sum of money will amount to £256. IO5. in 4 years at v^ per cent, simple interest ? £100 in 4 yrs. at 3^ per cent, amounts to £100 + (3i x4)£, or £114; and this £114 must be to the £256. IO5, as the £100 is to the required sum of money ; .-. £114 : £256 J- :: £100 : required number of pounds, whence, required number of pounds = £225, Ex. LIX. 1. What sum will amount to £150. 85. in 4 years at 5 per cent, simple interest ? 2. At what rate per cent, will £540 amount to £734. 8s. in 9 years, at simple interest ? 8. In what time will £350 amount to £402. 5fl. at 3 per cent, simple interest ? 4. At what rate per cent, will £325. 16?. 8^. amount to £374. Qs. Old. m S}. years, at simple interest ? 5. In what time will £142. 10.?. amount to £227- 5s. Od. at SI per cenf. simple interest? 6. At what rate will £157. 155. 4^. amount to £295. I65, Sd. in 25 years at simple interest? 7. What sum will produce for inter<3st £56. lis. in 2} years at 4} per cent, simple interest ? 8. What sum will amount to £105. 6s. Old. in 3i- years at 4|- per cent, simple interest ? COMPOUND INTEREST. 233 9. What sum will amount to £387. 7*. 7ld. in 3 years at 4 per cent., simple interest ? 10. In what time will £1375 amount to £1549. lis, at 8| per cent, simple interest ? 11. At what rate per cent., simple interest^ will £93G. IS^. 4c?. amount to £1157. 7*. Ud., in 4^^- years ? 12. In what time will £125 double itself at 5 per cent, simple interest ? 13. What sum will amount to £425. 19?. 4^d. in 10 years at 3.1- per cent, simple interest, and in how many more years w^ill it amount to £453. 11.?. 7^.? 14. What sum of principal money, lent out at 5 per cent, per annum, simple interest, will produce in 4 years the same amount of interest as £250, lent out at 3 per cent, per annum, will produce in G years ? COMPOUND INTEREST. 166. To find the Compound Interest of a given sum of money at a given rate per cent, for any number of years. Rule. " At the end of each year add the interest of that year, found by Art. (163), to the principal at the beginning of it ; this will be the principal for the next year ; proceed in the same way as far as may be required by the question. Add together the interests so arising in the several j'-ears, and the result will be the compound interest for the given period." The reason for the above Rule is clear from w4iat has been stated in Arts. (162 and 163). Ex. Required the compound interest and the amount of £720 for d years at 5 per cent. Proceeding as in Simple Interest for the 1'' year ; £720 5 £36-00 £720 = P' principal, by addition, £756 = 2"^ principal, of which find interest at 5 per cent. 5 £37-80 20 16-005. 234 ARITHMETIC, £756 =2"*^ principal, 37 . 16 = 2"'^ interest, £703 . 16 = 3"^ principal, of which find interest as above, 5 £39-69 . 20 13-80^. 12 9-60d. £793 . 16 . =^ principal for S'^'^ year, 39 . 13 . 91=:: interest for 3^^^ year, .*. £833 . 9 . 9f = amount of £720 in 3 years at 5 per cent. compound interest. The compound interest for that time = sum of interests for each year, = £36 + £37. 16^. + £39. 136\ 9fc^.=£113. 9^. 9f(/. JVote 1. It is customarj^ if the compound interest be required for anj'' number of entire years and a part of a year, (for instance for 5| years), to find the compound interest for the 6th year, and then take fths of the last interest for the fths of the 6th year. A^ote 2. If the interest be payable half-yearly, or quarterly, it is clear that the compound interest of a given sum for a given time will be greater as the length of each given period is less ; the simple interest will not be affected by the length of each period. A^te 3. As the vulgar fractions often in Compound Interest give considerable trouble, any sum in this Rule may be worked by means of decimals thus ; Ex. Find the amount of £625 at the end of 3 years at 4^ per cent. compound interest. 625 4-5 3] 25 2500 Principal for 1'* year £28125 £625 Int. for 1^' year £653-125 Principal for 2""^ year. COMPOUND INTEREST. 235 £658125 45 Principal for 2"^^ year 8265625 2612500 £29890625 £658125 Int for 2"^ year £682-515625 4-5 Principal for 8'''^ year 8412578125 2780062500 £80-718203125 £682-515625 Int. for 3"^ year £713-228828125 20 s.4-576562o00 12 d. 6-9187500 4 q. 8-67500 .-. Amount = £718. 4.^.6|. ...7 -f (6) £390 ...7 81 (7) i:572 ...8 ^ (8) £1261. Is. ... 1 l' (9) £35 ...4 4^ (10) £12.50 ...8 8.V (11) £2110 ...11 g' (12) £275. 65. 8c?. ...15 4 (13) £918 ... 4years 5 (14) £500 ...19 months 5} (15) 800gs. .. 20years 5^ (16) £2197 ... 3 years 4 compound interest 240 ARITHMETIC. 2. Find the Discount on (1) £63. Gs. 8d. due 4 months hence, at 4 per cent, per annum, [^simple interest. (2) £lSS0.7s.6d. ... 9 8 (3) £107.5^. ... 6 5 (4) £125. ]0^. ... 3 3i (5) £487 ... 5 si (6) £340 ... 5 4 (7) £3640 ... 10 4| (8) £813. 9^. ... Uyear 4J (9) £250. 15*. ...17 months 5 (10) £55 ...146 days 4| (11) A bill of £649 is dated on June 23, 1853, at 6 months, and is discounted on July 8, at 3^;- per cent. ; what does the banker gain thereby ? (12) Find the true discount on a bill drawn March 17, 1858, at 3 months, and discounted May 2, at 5f per cent. (13) Find the simple interest on £545 in 2 years, at 3^ per cent, per annum ; and the discount on £583. Ss. due 2 years hence, at the same rate of interest. Explain clearly why these two sums are identical. (14) Explain the difference between Discount and Interest. Five volumes of a work can be bought for a certain sum, payable at the end of a year ; and six volumes of the same work can be bought for the same sum in ready money : what is the rate of discount ? (15) A tradesman marks his goods with two prices, one for ready money, and the other for one year's credit allowing discount at 5 per cent. ; if the credit price be marked at £2. 9*., what ought to be the cash price ? STOCKS. 170. If the 8 per cent, consols be quoted in the money-market at 96|, the meaning of this is, that for £90. 7s. 6d. of money a person can purchase £100 stock, for which he will receive an acknowledgment which will entitle him to half-yearly dividends from Government, at the rate of 8 per cent, per annum on the stock held by him. Similarly, if shares in any trading company, which were originally fixed at any given amount, say £100 each, be advertised in the share- STOCKS. 241 Aiarket at 86, the meaning of this is, that for £86 of money one share can be obtained, and the holder of such share will receive dividends at the end of each half-year upon the £100 share, according to the state of the hnances of the company. Def. Stock may therefore be defined to be the capital of trading companies ; or to be the money borrowed by our or any other Govern- ment, at so much per cent., to defray the expenses of the nation. The amount of debt owing by the Government is called the National Debt^ or the Funds. Tlie Funds represent the credit of the country, which is bound to pay whatever debts are contracted by its Government. The government, however, reserves to itself the option of paying off the principal at any future time whatever ; pledging itself, nevertheless, to pay the interest on it regularly at fixed periods, in the mean time. From a variety of causes the price of stock is continually varying. A fundholder can at any time convert his stock into money, and it will de- pend upon the price at which he disposes of his stock, as compared with that at which he bought it, whether he will gain or lose by the trans- action. Note 1. Purchases or sales of stock are generally made through Brokers, who charge £|, or 2*. Qd., per cent, upon the stock bought or sold : so that in practice, wdien stock is bought by any party, every £100 stock costs that party £^ more than the market-price of the stock : and v.-hen stock is sold, the seller gets £^ less for every £100 stock sold than the market-price. Thus, the actual cost of £100 stock in the 8 per cents, at 94Jr, is £(94^ + ^), or £04|. The actual sum received for £100 stock in the S per cents, at 941 jg £(9^1 _i), or £94. Unless the brokerage is mentioned, it need not be noticed in working examples in stocks. Note 2. When the price of £100 stock is £100 in money, the stock is said to be at pcz>*. ^Fhen the price of £100 stock is more than £100 in money, the stock is said to be at a. premium. When the price of £100 stock is less than £100 in money, the stock is said to be at a discount. All examples in Stocks depend on the principles of propjortion : those of most frequent occurrence will he now explained. Ex. 1 . Required the sum which will purchase £1500 in the 3 per cents, at 82. 16 242 ARITHMETIC. In this case £100 stock costs £82 in money ; /. £100 stock : £1500 stock :: £82 money : required sum of money; whence, required sum of money = £1230. Ex. 2. What amount of stock in the 3} per cents, at 00 will £4050 [iurchase ? In this case £90 money will purchase £100 stock ; .*. £90 : £4050 :: £100 stock : required amount of stock j whence, required amount of stock - £4500. Ex. 3. If I huy £1520 3 per cent, consols at 98}, and pay £|- for brokerage, Avhat does it cost me ? Every £100 stock costs me £(93J + ^), or £93| ; .*. £100 stock : £1520 stock :: £93| : required sum of money; whence, required sum of money — £1419. Qs. Ex. 4. "What sterling money sliall I receive for £1920. 13s. M. in the OJ per cents, at 98 J, brokerage being £^- per cent. ? £100 stock realizes £(98|-i)^£982- ; ,*. £100 stock : £1920| stock :: £98f- : required sterling money; whence, required sterling money = £1896. 13^. 2c?. Ex. 5. If I invest £7927. 10.9. in the 3 per cents, at 94|, what annual income siiall I receive from the investment ? For every £94| I get £100 stock, and the interest on £100 stock is £8 ; therefore for every £94f of money I get £3 interest ; .*. £94| : £7927. 10.?. :: £3 : required annual income; whence, required annual income = £252. Note 3. If it be required to find the income arising from a certain quantity of stock, it is merely a question of simple interest. A^te 4. It may be noticed in the above examples, that when the question was simply to find amount of stock, or money realized by sale of stock, the 3, 4, or other rate per cent, never entered into the statement; and when the question was simply to find income arising from any sum invested in the funds, then the £100 never entered into the statement. Ex. 6. Which is the best stock to invest £1000 in, the 3 per cents. at 89|-, or the 3.} per cents, at 98J ? In the first case, every £89| of money gives £8 interest ; 3 6 .*. every £1 of money gives £— -: , or £f^ interest. STOCKS. 243 In the second case, every £98^ of money gives £3} interest ; S}- 7 .'. every £1 of money gives £-j^^- , or £^-^, interest ; 6 7 and comparing the fractions —-, and -,^ , ^ ° 179 197 since 7x]79 Is>Gxl97, the 2"*^ fraction is greater than the 1''^ and therefore the 2"*^ investment the hest. Ex. 7. How much stock can he purchased by the transfer of £2000 stock from the 3 per cents, at 90 to the 3.} per cents, at 9G ; and what change will be effected in income by it ? In order to find how much stock at 96 can he purchased for £2000 stock at 90, we must consider that the liigher the price of the stock the less will the quantity of it produced be by the purchase, so that we must state as follows ; 96 ; 90 :: £2000 stock : required amount of stock, whence, required amount of stock = £1875. Income in first case = £60, income in second case = £65. 125. 6d.; .'. income is increased by £5. 12*. 6d. Note 5. All questions of the transfer of stock from one kind to another, belong to the Rule of Three Inverse. Note 6. The last question might have been worked thus : first sell out the stock at 90, and then invest the proceeds in 3A- per cents, at QQ. Ex. 8. A person purchases £1000 3 per cent, consols at 97J, and sell? out again when they have sunk to 83J ; how much does he lose by the transaction ? He loses on every £100 stock £(97^- - 83-?r), or £13f ; .•. his total loss = £(13f x 10) =^£136. 6s. Ex. LXII. 1. Find the quantity of stock purchased by investing : (1) £2850 in the 3 per cents, at 75. (2) £712 in the 3i per cents, at 89. (8) £504 in the 4 per cents, at 96. (4) £883. 5 fl. in the 4 per cents, at 93. 16-2 244 ARITHMETIC. (5) £3741 in the 8^ per cents, at 87. (6) X'500 in the 3 per cents, at 83|. (7) £800 in the 4 per cents, at 75^. (8) £4311. 8s. 9d. in the 3^ per cents, at 85|. (9) 2000 guineas in the 3^- per cents, at 94. (10) £23.53 in the 3 per cents, at 90|, brokerage ^ per cent. (11) £3277 in the 4 per cents, at 105|-, brokerage ^ per cent. (12) 10000 guineas in the 3^ per cents, at 99}, brokerage |^ per cent. 2. Find the value in sterling monc}' of (1) £2G00 in the 4 per cents, at 93. (2) £1920 in the 3 per cents, at 77 J. (3) £3000 in the 3| per cents, at 921 (4) £2240 in the 3} per cents, at 81 1-. (5) £3416 3 per cent, stock at 89 per cent. (6) £1743 3} per cent, stock at 82|- per cent. (7) £2675 4 per cent, stock at 91f per cent. (8) £1000 4 per cent, stock at 97f per cent., brokerage ^ per cent. (9) £2153. 10*. bank stock at 188f per cent., brokerage J per cent. 8. Find the yearly income arising from the investment of (1) £1008 in the 3 per cents, at 84. (2) £5580 in the 4 per cents, at 93. (3) £1138. 5 fi, in the U per cents, at 92. (4) £1638 in the 4^^ per cents, at 93|. (5) £2000 in the 3 per cents, at 88^. (6) £3425. 15*. 2d. in the 3 per cents, at 91|. (7) £4788 in the 81 per cents, at 105. (8) £3500 in the 3 per cent, consols at 94^, brokerage ^ per cent. (9) 5000 guineas in the 3J per cents, at 102|, brokerage J per cent. 4. What sums of money must be invested in the undermentioned stocks in order to produce the following incomes ? (1) £60 in the 8 per cents, at 85. (2) £288 in the 3 per cents, at 67. (3) £70 in the 31- per cents, at 90. (4) £83. 2 fl. 5 c. in the 4i- per cents, at 94. STOCKS. 245 (5) i'87 in the 3 per cents, cat 74^, brokerage |- per cent. (6) £37. 10*. in the 4 per cents, at 931, brokerage ^ per cent. 5. At what rate per cent. Avill a person receive interest who invests his capital ? (1) In the 8 per cents, at 01. (2) In the 3^ per cents, at 94. (3) In the 4^ per cents, at 96|, brokerage | per cent. (4) In the 5 per cents, at 102|r, brokerage } per cent. 6. If £7927. 10*. be laid out in purchasing 3 per cent, stock at 94^, ■what annual income will be derived from this investment, after deducting an income-tax of 7d. in the pound ? 7. A person invested money in the 3 per cent, consols when they were at 90, and some more when they w^ere at 80 ; find the rate of interest he obtained in each case, and the advantage per cent, of the second purchase over the first. 8. Find the income which will be derived from a capital of £2000, if f ths of it be invested in the 3 per cents, at 98, and the remainder in the 3| per cents, at par. 9. If a person receives 4.^ per cent, interest on his capital by investing in the 3^ per cents., what is the price of the stock, and how much stock can be purchased for £1200 ? 10. How much money must a broker invest in the funds when con- sols are at 90, so as to procure the same income as if he had invested £1100 W'hen consols were at 99 ? 11. A person buys £500 stock at 98|, and sells out at 103 ; what does he gain by the transaction ? 12. A person invests 9000 guineas in the 3 per cents, at 81, and sells out when they have sunk to 67^ ; how much does he lose by the trans- action ? 13. When £100 stock may be purchased in the 3 per cents, for £89^, at what rate may the same quantity of stock be purchased in the 8^ per cents, with equal advantage 1 14. A person invests his share of a legacy of £1000, which is a third, in the 3 per cents, at 88| per cent., find his half-yearly dividends. 15. A person transfers £1000 stock from the 4 per cents, at 90, to the 3 per cents, at 72 ; find the alteration in his income. 16. What incomes will £5000 of 8^ per cent, stock, and £5000 sterling invested in the 3^- per cent, stock at 102|, respectively pro« duce ? 246 ARITHMETIC. 17. Find the income produced bj'- £12600 of 3 per cent, stock; and its sterling value, when the stocks are at 95. 18. A person transfers £3000 stock from the 3 per cent, consols at 89|, to the reduced 3i- per cents, at 98^ : find what quantity of the latter he will hold, and the alteration in his income. 19. Which is the best stock to invest £10000 in, the 8 per cents, at 90| , or the 4 per cents at 101 ? 20. A person invests £1087. 10*. in the 3 per cents, at 83, and when the funds have risen 1 per cent, lie transfers his capital to the 4 per cents, at 96 : find the alteration in his income. 21. Which is the better investment, the 8h per cents, at 96, or the 4 per cents, at 111, and what is the difference per cent, between them ? 22. If £512 be invested in the 3 per cents, at 96, what will be the half-yearly interest, after deducting an income-tax of 7c?. in the pound? 23. How much in the 3 per cents, at 96 must be sold out to pay a bill of £1654, 9 months before it becomes due, real discount being allowed at 4^ per cent, per annum ? 24. Which is the better investment, £1896 in the Sh per cents, at 87, or in railway shares at £89 per share, the dividends in the latter case behig 8| per cent, on the sum invested ? 25. A person has £2950 in the 3 per cents, at 83^ ; when the funds have Mien 2^ per cent., he transfers his capital into the 5 per cents, at 108; find the alteration in his income. 26. Which would be the best investment, 3 per cent, stock at 87|, or shares at £233 each, on each of which a dividend of £7. 13*. M. is paid annually? What sum must be invested in the former to produce an annual income of £460 ? and what in the latter ? 27. If the 3^ per cents, be at 91, how much must a person invest in order that he may have a yearly income of £460, after paying 7d. in the pound for income-tax ? 28. The dividends on a certain amount of 3 per cent, stock accumu- lated in 13 years to £3081. How much stock was there, and what will it be worth if the stock be sold at 79| ? 29. A person possesses £3200 3 per cents., which he sells at 99| : he invests the proceeds in railway shares at £56 a share, which shares pay 5 per cent, interest on £45, the amount paid on each share. How much is his income altered by the transaction ? SO. If I lay out £1911 in the purchase of 3 per cent, consols, when they are at 70:.', at what price should I sell out my stock again in order I PROFIT AND LOSS. 247 to realize on the whole a gain of £150, after having paid ^rth per cent, fur commission on each transaction? 31. A person had X10,000 in the 8 per cent. South Sea Annuities, and the Government offered to give £110 hearing interest at the rate of 2S per cent, for every £100 of these annuities, or to pay the £10,000 in cash on a certain day. The latter proposal was preferred, and on the money being paid it was re-invested in consols at 93 How much would he have lost in income had. he accepted the first proposal, and what will he now gain hy the new investment? 32. AVhat sum would be saved annually if the interest on a public debt of £4,000,000 were reduced from 3.^ per cent, to 3 per cent. ? If in consequence the price of this stock fell from £101 to £95f:, how much would ths whole property of the fundholders be diminished ? PEOFIT AND LOSS. 171. Dep. All questions in Arithmetic which relate to gain or loss in mercantile transactions, fall under the head of Profit and Loss. Examples in Profit and Loss are luorked by the principle of Proportion : various examples will now he worked out hy way of illustration. Ex. 1. If a cask of wine containing 84 gallons cost £112. 6s. y what is gained by selling it at 31*. Qd. per gallon? The gain = selling price less first cost; the selling price = (3U x 84)5. ^£132. Qs.; therefore the gain = £132. 6^. -£112. 5*. = £20. Is. Ex. 2. A ream of paper cost me 21*. Qd., what must I sell it at, so r^s to realize 20 per cent. ? The reasoning in this case is, If £100 gain £20, or produce £120, what will 21*. Qd. produce ? .*. £100 : 21s. Gd. :: £120 : required amount in pounds, whence, required amount = £1. 5*. 9'jd. Ex. 3. If I buy hay at £4. 10*. a ton, what must I sell it at to lose 15 per cent. ? In this case, every £100 would realize £(100 — 15), or £85 ; .'. £100 : £4. 16i\ :: £85 : required amount in pound;:^, whence, required amount = £4. 1*. 7^^^ 248 ARITHMETIC. Ex. 4. A man buys 33 geese for £10; at how mucli per head must he sell them to gain 10 per cent, on his outlay 1 Im this case, £100 : £10 :: £110 : selling price of the geese in pounds, whence, selling price = £11, .". selling price of each goose = £^^ = Gs. Sd. Ex. 5. A person buys shares in a railway when they are at £19i-, £15 having been paid, and sells them at £32. 9*. when £25 has been paid : how much per cent, does he gain? He buys eacli siiare at £19i^, and he afterwards pays upon it £(25 — 15), or £10; therefore at the time he sells, he has paid on each share £29. 10.?.; therefore by selling at £32. 9.?. he gains on each £29. 10*. which he has paid (£82. 9,5. -£29^ 10.^.) = £2. 19s. ; .'. £2dh : £100 :: £2i^§ : gain per cent in pounds ; whence, gain per cent. = £10, or gain is 10 per cent. Ex. 6. What was the prime cost of an article, which when sold for 125-., realized a profit of 20 per cent.? Here what cost £100 would be sold for £120 ; .-. £120 : 12s. :: £100 : prime cost in pounds, whence, prime cost = £^ = 10^. If the above example had been, " What was the prime cost of an article, which when sold for 126-., entails a loss of 20 per cent.?" then £80 : 12.^. :: £100 : prime cost in pounds, whence, prime cost = £j = 15*, Ex. 7. If by selling a horse for £40 I lose 20 per cent., what must I have sold him for so as to gain 10 per cent.? Here what would cost me £100 must be sold in one case for £80, and in the other for £110; and therefore we get this statement; selling price of £100 in 1'* case : selling price of horse in 1'' case :: selling price of £100 in 2""* case : selling price of horse in 2"'^ case ; or £80 : £40 :: £110 : selling price in pounds; whence, selling price = £55. Ex. 8. A grocer buys 8 cwt. of sugar at Gd. a lb., 2 cwt. of sugar at lO^c?. a lb., and 21 qrs. of sugar at 1.?. a lb. ; and mixes them : he sells 4 cwt. of the mixture at 9c?. a lb. What must he sell the remainder at, in order to gain 25 per cent, on his outlay ? PROFIT AND LOSS. 249 £. s. d. 8 cwt., or 836lbs., at 6d. a lb , cost 8. 8.0 2 cwt., or 224 lbs., at lOld. a lb., cost 9 . 16 . 2.}qrs., or 701bs., at 1*. a lb., cost 3 . 10 . .-. 630 lbs. cost . . 21 . 14 . In order to gain 25 per cent, on £21. 14^^., it must realize £27. 25. 6d. ; £. s. d. .'. he must sell 630 lbs. for ... 27 . 2.6 but he sells 448 lbs. for ... 16 . 16 . .-. by Subt" he must sell 182 lbs. for ... 10 . G. 6 /. he must sell lib. for — — ^— ^^ ' , or IS Ad. 182 ^"^ Ex. LXIII. 1. Bought 5 cwt. 3 qrs. 14 lbs. of cheese at £1. 12*. per cwt., and sold it again for £2. Oj. 8d. per cwt. What was the gain upon the whole ? 2. If 5 cwt. 3 qrs. 14 lbs. be bought for £9. 8*. and sold for £11. 186\ lid. what is the rate of gain per cwt. ? 3. Find the total value of 43 articles at £4. Qs. 8d. each, 57 at £11. Ss. 6d. each, and 4 at £13. 1.5.?. 4(i. each. What is gained or lost by selling them at the rate of 8 for £28 ? 4. A person buys 400 yards of silk at £80, and sells 800 yards at 5s. 6d. a yard, and the rest, which is damaged, at 25. a yard; find how much per cent, he gains or loses. 5. A grocer buys 2 cwt. of sugar at Gd. per pound, and 4 cwt. at 4^(/.; he sells 8 cwt. at 5id- per pound ; at what rate per pound will he be able to sell tlie remainder so as neither to gain nor lose by the bargain ? 6. If a commodity be bought for £3. 8s. 5d. a cwt. and sold for 8d. a lb., find the rate of profit per cent. 7. Bought goods at Gtd. per pound, and sold them at £4. 10^. per c^vt.; what is the gain or loss per cent. ? 8. An article which cost 3*. Gd. is sold for Ss. lO^of. ; find the gain per cent. 9. Goods -were sold at 12 guineas, at a profit of 22 Jj per cent.; what was the prime cost ? 10. If a tradesman gain 5*. Gd. on an article which he sells for 22*. what is his gain per cent. ? 11. A man sells a horse for £24. 12^., and loses £18. per cent, on what the horse cost him ; what was the original cost ? 250 ARITHMETIC. 12. By selling an article for 5s. a person loses 5 per cent. ; what was the prime cost, and what must he sell it at to gain 4^; per cent. ? 13. The cost price of a book is 6,9. 8d.; the expense of sale 5 per cent. upon the cost price ; and the profit 25 per cent, upon the whole outlay : find the selling price of the book. 14. If by selling an article for £25. 10,y. 8 per cent, be lost, what per cent, is gained or lost if it be sold at £38 ? 15. I bought 500 sheep at £2. 2s a-head ; their food cost me 5s. Qd. a-head : I then sold them at £2. 8s. Gd. a-head. Find my whole gain, and also my gain per cent. 16. A person having bought goods for £-10 sells half of them at a gain of 5 per cent. ; for how much must he sell the remainder so as to gain 20 per cent, on the whole ? 17. A vintner buys a cask of wine containing 36 gallons at 10^. per gallon ; he keeps it for four years, and then finds that he has lost 6 gallons by leakage ; at what price per gallon must he sell the remainder in order that he may realize 20 per cent, upon his outlay ? 18. A person rents a piece of land for £120 a year. He lays out £625 in buying 50 bullocks. At the end of the year he sells thcmj having expended £12. lOs. in labour. How much per head must he gain by them in order to realize his rent and expenses, and 10 per cent, upon his original outlay ? 19. A grocer mixes two kinds of tea which cost him 3.?. 8d. and 4^. M. per lb. respectively ; what must be the selling price of the mixture in order that he may gain 15 per cent, on his outlay ? 20. A person has goods worth £80 ; he sells one-third of them so as to lose 10 per cent. ; what must he sell the remainder at so as to gain 20 per cent, on the whole ? 21. I buy a house for 500 guineas, and sell it immediately at a profit of 30 per cent. ; what do I receive, supposing the expenses of the sale to be 5 per cent. 1 22. The prime cost of a 76-gallon cask is £23. 12s. 6d., but 13 gallons are lost by leakage ; 9 gallons of Avater is then mixed with the remainder, and it is sold at 7^. Gd. a gallon. Find the whole gain, and also the gain per cent. 23. A stationer sold quills at 11*. a thousand, by which he cleared | of the money ; he raises the price to 13^. 6^?. What does he clear per cent, by the latter price 1 24. A person sold 72 yards of cloth for £8. 14^. ; his profit being the cost of 11*52 yards : how much did lie gain per cont. 1 DIVISION INTO PROPORTIONAL PARTS. 251 25. A smuggler buys 6 cwt. of tobacco at 1*. Sd. per lb. ; he meets with a revenue-officer, who seizes Ird of it : at what rate per lb. must he sell the remainder, so as, 1st, neither to gain or lose; 2nd, to gain 5 guineas ; and Srd, to gain cent, per cent. ? 26. A person expends £3000 in railway shares at 15^ per cent, dis- count, and sells them at i^ar ; what does he gain by the transaction, and ■what per cent. ? 27. A wine-merchant bought 14;. pipes of wine, which having received damage, he sold for £1120^1^, thereby losing 20 per cent. ; find the cost of the wine per pipe, and the selling price of it per gallon. 28. A farm is let for £96 and the value of a certain number of quar- ters of wlieat. When wheat is 38s. a quarter, the whole rent is 15 per cent, lower than when it is 56*. a quarter. Find the number of quarters of wheat which are paid as part of the rent. 29. A man having bought a lot of goods for £150, sells |Vrd at a loss of 4 per cent. ; by Avhat increase per cent, must he raise that selling price, in order that by selling tlie rest at the increased rate, he may gain 4 per cent, on the whole transaction 1 SO. A person bought a French watch, bearing a duty of 25 per cent., and sold it at a loss of 5 per cent. ; had he sold it for £3 more, he w^ould have cleared 1 per cent, on his bargain. What had the French maker for it ? DIVISION INTO P.ROPOETIONAL PAKTS. 172. To divide a given numher into parts which shall he proportional to certain other given number's. This is merely an application of the Rule of Three ; still it may be well to state a general Rule, by which examples which come under the above head may be worked. Rule. State thus : " As the sum of the given parts : any one of them :: the entire quantity to be divided : the corresponding part of it." This statement must be repeated for each of the parts, or at all events for all but the last part, which of course may either be found by the Rule, or by subtracting the sum of the values of the other parts from the entire quantity to be divided. Ex. 1. Divide 40 guineas among A, B, and C, so that their por- tions may be as 7, 11, and 14 respectively. 252 ARITHMETIC. Proceeding according to the Rule given above, 32 : 7 :: 40 guineas : A's share, 32 : 11 :: 40 guineas : B's share, whence ^'s share = £9. Ss. Od., and B's share = £14. 8^. 9d. C's share may he found from the proportion 32 : 14 :: 40 guineas : C's share ; whence C's share = £18. 7*. (yd. ; or by subtracting £9. 8s. 9o?. + £l4. 8s. 9d., or £23. 12*. 6d. from £42, which leaves £18. 7*. Gd., as above. The reason for the above process is clear from the consideration, that 40 guineas is to be divided into 32 equal parts, of which A is to have 7 parts, B 11, and C 14. Ex. 2. Divide £11000 among 4 persons, A, B, C, D, in the propor- tions of 1 ^, 1, and |. Sum of shares = ^; '"' 60^ • 3 •• -£11000 : A's share in pounds, whence A's share = £4285. 14*. 3^d. Similarly, B's share = £2857. 2.s. 10 fd., C's share = £2142. 17*. Ifc?. D's share = £1714. 5s. 8^d. Ex. 3. Divide £45000 among A, B, C, and D, so that A's share : B's share :: 1 : 2, B's : C's :: 3 : 4, and C's : D's :: 4 : 5. In this case, B's share = 2 ^'s share, 8 C's share — 4 B's share, 4 D's share = 5 C's share ; .'. we have C's share = 4 B's share = § A's share, and D's share = ^ C's share = ^^ A's share ; .'. ^'s share + B's share + C's share + D's share ^ A's share x (1 + 2 + 1 + V"), = 9 ^'s share ; .-. A's share = £5000, i?'s = £10000, C's ^£18338. Qs. 8d., J)'s-- £16666. 13*. 4f/. FELLOWSHIP OR PARTNERSHIP. 253 FELLOWSHIP OR PARTNERSHIP. 173. Def. Fellowship or Partnership is a method by whicli the respective gains or losses of i)artners m any mercantile transactions are determined. Fellowship is divided into Simple and Compound Fellowship : in the former, the sums of money put in by the several partners continue in the business for the same time ; in the latter, for different periods of time. SIMPLE FELLOWSHIP. 174. Examples in this Rule are merely particular applications of the Rule in Art. (172), and that Rule therefore applies. Ex. 1. Two merchants,^ and B, form a joint capital; J puts in £240, and B £360 : they gain £80. How ought the gain to be divided between them ? £(240 + 860) : £240 :: £80 : ^'s share in pounds, whence, J's share = £32, and .*. ^'s share = £48. Note. The estate of a Bankrupt may be divided among his creditors by the same method. Ex. 2. A bankrupt owes three creditors. A, B, and C, £175, £210, and £265, respectively ; his property is worth £422. IO5. : what ought they each to receive ? £650 : £175 :: £422| : .4's share, £650 : £210 :: £422.1 : J5's share, \vhence A'& share = £118. lo^., B'?> share = £186. IO5. ; .-. C's share = £172. 6s. COMPOUND FELLOWSHIP. 175. Rule. "Reduce all the times into the same denomination, and multiply each man's stock by the time of its continuance, and then state thus : As the sum of all the products : each particular product :: the whole quantity to be divided : the corresponding share." 254 AEITHMETIC. Ex. 1. A and B enter into partnership ; A contributes £3000 for 9 months, and B £2400 for 6 months, they gain £1150 : find each man's share of the gain. Proceeding by the Rule given above, £(3000 X 9 + 2400 x G) : £(3000 x 9) :: £1150 : ^'s share of gain, or £41400 : £27000 :: £1150 : A's sliare of gain, and £41400 : £14400 :: £1150 : B's share of gain ; whence, ^'s share = £750, and B's share = £400. The reason for the above process is evident from the consideration, that a stock of £3000 for 9 months would be equivalent to a stock of 9 times £3000 for 1 month ; and one of £2400 for 6 months, to one of 6 times £2400 for 1 month : hence, the increased stocks being considered, the question then becomes one of Simple Fellowship. Ex. 2. There were at a feast 20 men, 30 women, and 15 servants; for every 106^. that a man paid, a woman paid Qs., and a servant 2*. ; the bill amounted to £41 : how much aid each man, woman, and servant pay? 20 men at 10^. each = 200 at \s., 80 women at 65. = 180 at 1^., and 15 servants at 2*. = 80 at Is. ; and 200 + 180 + 80 = 410. Hence we have 410 : 200 :: £41 : 20 men's share (in pounds) ; 410 : 180 :: £41 : 80 women's share (in pounds) ; 410 : 80 :: £41 : 15 servants' share (in pounds); .*. 20 men's shares =£20, 30 women's shares = £18, and 15 servants' shares = £3; /. each man paid £1, each woman 12*., and each servant 4.5. EQUATION OF PAYMENTS. 176. Def. "IVlien a person owes another several sums of money, due at different times, the Rule by which we determine the just time when the whole debt may be discharged at one payment, is called the Equation of Payments. Note. It is assumed in this Rule that the sum of the interests of the several debts for their respective times equals the interest of the sum of the debts for the equated time. Rule. " Multiply each debt into the time which will elapse before it becomes due, and then divide the sum of the products by the sum of the debts ; the quotient will be the equated time required."' EQUATION OF PAYMENTS. 255 Ex. 1. A owes B £100, whereof £40 is to be paid in 3 months, and £G0 in 5 months : find the equated time. Proceeding according to the Rule given above, then (40 X 3 + 60 X 5) = (40 + 60) x equated time in months, whence, equated time = 4i months. T7ie reason for the above process, in accordance with our assumption, is clear from the consideration that the sum of the interests of £40 for 3 months, and £60 for 5 months, is the same as the interest of £(120 + 300), or £420 for 1 month ; if therefore A has to pay £100 in one sum, the question is, how long ought he to hold it so tliat the interest on it may be the same as the interest on £420 for 1 month. The statement therefore will be this : £100 : £420 :: 1 month : required number of months ; whence, required number of months = 4i months ; which is evidently the equated time of payment, and agrees with the result obtained by the Rule given above. Ex. 2. A owed B £1000, to be paid at the end of 9 months ; he pays however £200 at tlie end of 3 months, and £300 at the end of 8 months : when was the remainder due ? In this case, (200 X 3 + 300 X 8 + 500 X number of months required) = 1000 x 9, or 500 X number of months required == GOOO ; whence number of months required = 12. Ex. LXIV. 1. A company of militia consisting of 72 men is to be raised from 3 towns, which contain respectively 15C0, 7000, and 9500 men. How many must each town provide ? 2. Divide £17. H^- ^d. into two parts which shall be to each other as 5 : 16. 3. Divide 4472 into parts which shall be to each other in the ratio of 8, 5, 7, 11 ; and also £500 into parts which shall be in the ratio of ^, f, and -f . 4. A bankrupt owes A £256. Qs. Qd., B £203. 10.5., and C £141. 13^. M.; his estate is worth £421. l^-. ; how much will A, B, and C receive respectively? ,i-': r 5. A mass of counterfeit metal is composed of fine gold 15 parts, silver 4 parts, and copper 3 parts : find how much of each is required in making 18 cwt. of the composition. 256 ARITHMETIC. 6. Two persons have gained in trade £720; the one put in £2200 and the otlier £1800; what is each person's share of the profits? 7. In a certain substance there are 11 parts tin to 100 of copper. Find the weight of tin in a piece weighing 24 cwt. 1 8. A man leaves his property amounting to £13,000 to be divided amongst his children, consisting of 4 sons and 3 daughters ; the three younger sons are each to liave twice the share of each of the daughters, and the eldest son as much as a younger son and a daughter together ; find the sliare of each. 9. Two persons, A and B, are partners in a mercantile concern, and contribute £1200 and £2000 capital respectively ; ^ is to have 10 per cent, of the profits for managing the business, and the remaining profits to be divided in proportion to the capital contributed by each ; the entire profit at the year's end is £800; how much of it must each receive? 10. Divide £100 among A, B, C, and D, so that B may receive as much as ^ ; C as much as A and B together ; and D as much as A, B^ and C together. 11. Divide £11,875 among A, B, and C, so that as often as A gets £4, B shall get £3, and as often as B gets £6, C shall get £5. 12. A commences business with a capital of £1000, two years after- wards he takes B into partnership with a capital of £15,000, and in 3 years more they divide a profit of £1500 ; required 5 s share. 13. £700 is due in 3 months, £800 in 5 months, and £500 in 10 months ; find the equated time of payment. 14. Find the equated time of payment of £750, one half of which is due in 4 months, f in 5 months, and the rest in 6 months. 15. A owes B a debt payable in 7g'o months, but he paj^s g- in 4 months, |- in G months, ^ in 8 months ; when ought the remainder to be paid ? 16. A, B, and C rent a field for £11. Qs. ; A puts in 70 cattle for 6 months ; B 40 for 9 months ; and C50 for 7 months ; what ought C to pay ? 17. A, B, and C invest capital to the amount of £700, £500, and £800 respectively ; A was to have 25 per cent, of the profits, which amount to £450 ; what share of the profits ought C to have ? 18. A and B enter into a speculation ; A puts in £50 and B puts in £45 ; at the end of 4 months A withdraws i his capital, and at the end of G months B withdraws ^ of his; C then enters with a capital of £70; at the end of 12 months their profits are £254; how ought this to be divided amongst them ? APPLICATIONS OF THE TERM PER CENT. 257 APPLICATIONS OF THE TERM PER CENT. 177. In Art. 162, and those which follow, wherever the term " Per Cent." occurred it referred to £100 money, or £100 stock; there are however many cases in which the term Per Cent, occurs, where the re- ference is neither to the one nor the other, but to the number 100, where the unit is an abstract number, or a concrete number of a different kind from the above mentioned. All such examples depend on the principles of proportion : some ex- amples will now be worked by way of illustration, and others subjoined for practice. Ex. 1. Find how much per cent. 7 is of 16 1 In other words the question is ; find what number bears the same ratio to 100, that 7 bears to 16. By Rule, Art. 155, 16 : 100 :: 7 : number required ; .*. number required = — - = 4o* t o. Ex. 2. In a parish school of 153 children, 125 learn to write. Whab is the percentage ? In other words, what number bears the same ratio to 100, which 125 bears to 153? .-. 153 : 100 :: 125 ; percentage; 12500 „J07 ••• P^^^^^^^Se = -i^ = 81 — . Ex. 3. In 1842 the number of the members of the University of Cambridge was 5853, and in 1852 the number was 6397, find the increase per cent. Subtracting 5853 from 6397 we obtain 544 the increase on 5853 mem- bers; the question then is this; If 5853 members give an increase of 544, what increase do 100 members give? .*. 5853 : 100 :: 544 : increase per cent.; , 54400 ^1723 .-. mcrease per cent.^^^^^g =9^^3. Ex. 4. 23 per cent, of the population of a town containing 80000 people died of cholera ; find the number of deaths. 17 258 ARITHMETIC. If 23 died out of every 100, how many died out of 30000 ? 100 : 30000 :: 23 : number of deaths j .'. number of deaths^— r^— ==G900. Ex. 5. Between the years 1821 and 183] the populajtion of Norwich increased by 22 per cent., and in the latter year it was G1116. "What was it in 1821 ? For every 122 persons in 1831 there were 100 persons in 1821 ; .*. 122 : 61116 :: 100 : number required ; , . , 61116x100 __„. , .-. number required = —^ =50095 nearly. Ex. 6. If of a regiment of 750 men, 26 per cent, are in hospital, 82 per cent, in trenches, and the rest in camp, how many are in hospital, trenches, and camp respectively ? 100 : 750 :: 26 : number in hospital; .-. number in hospital^ — — — — =195. 100 : 750 :: 32 : number in trenches; .-. number in trenches = — t7)K~ -240; .-. number in camp = 750 -(195 + 240) = 315. Ex. 7. The percentage of children who are learning to write is 65 in a school of 60 children, and 78 in another school of 70, what is the percentage in the two schools together ? In the first school, 100 : 60 :: 65 : number v.'ho learn to write; .*. number v.'ho learn to write = — ttjx— — 39. In the second school, 100 : 70 :: 78 : number who learn to write; .'. number who learn to write = ^-r^.— =54| ; ,% in a school of (60 + 70) or of 130, there are 93| who learn to write ; ,'. 130 : 100 :: 93f : percentage required; . , 100x98f ^^ /. percentage required = — fso~~ APPLICATIONS OF THE TERM PER CENT. 259 Ex. 8. In standard gold 11 parts out of 12 are pure gold ; how much per cent, is dross ? In every 12 parts 1 part is dross, .*. 12 : 100 ;: 1 : percentage of dross; .*. percentage of dross == -^^ - 8^. Ex. 9. Archimedes discovered that the crown made for King Hiero consisted of gold and silver in the ratio of 2 : 1. How much per cent. Was gold, and how much per cent, silver ? Out of every 8 parts, 2 were gold, and 1 silver ; .-. 3 : 100 :: 2 : percentage of gold; - ,- 100x2 .*. percentage of gold = — -^^ — — bog ; o and percentage of silver = 83 J. Ex. 10. The numbers of male and female criminals are 1235 and 988 respectively; while the decrease in the former is 4'6 per cent., the increase in the latter is 9'8 per cent. ; find the increase or decrease per cent, in the whole number of crimmals. 1st. 100 ; 1285 :: 4*6 : whole decrease of male criminals ; /. whole decrease of male criminals — — — -- — =56*81. 2nd. 100 : 988 :: 9"8 . whole increase of female criminals ; /. whole increase of female criminals = — , =96*824; .'. in (1235 + 988) or 2223 persons there is an increase of (96-824 -56-81) or 40-014 pei-sons. /, 2223 : 100 :: 40014 : percentage required ; . , 4001-4 ^ _ .'. percentage requn-ed = -90^^ =18. Ex. LXV. 1. What is the percentage on 56394 at ^ ; § ; 4 ; 7^ ; 10; 150^ i 2. How much per cent, is 15 of 96 ; 19 of 81 ; 23 of 256 ; 185i of 7821-75 ; 5-3 of 11080-5 ? 8. Write in a decimal form J ; 2J ; 4J ; 5^ ; 26^ ; 230-05 ; 500-0138 per cent. 17—2 260 ARITHMETIC. 4. A cask, which contained 2005 gallons, leaked 27 per cent., how much remained in the cask? 5. A malster malts 7500 bushels of barley, which in the process increases 12.} per cent., how many bushels of malt has he ? 6. A grocer uses for a 1 lb. weight one which only weighs 15-75 oz., what does he gain per cent, by his dishonesty "? 7. Out of 14804 cases of Small- Pox 1588 persons died, and out of 2422 cases of Scarlet Fever 211 persons died; find the rate per cent. of mortality in each case, also the rate per cent, of mortality in the whole number of sick people. 8. The population of Ireland was 7767401 in 1831, 8175124 in 1841, 6515794 in 1851. Find the increase per cent, in the first ten years, the decrease per cent, in the second ten years, and the decrease per cent, in the 20 years from 1831 to 1851. 9. The population of a city is a million ; it rises 1|- per cent, for 3 years successively ; find the population at the end of 3 years. 10. A school contains 383 scholars, 3 are of the age of 18 years ; 5 per cent, of the remainder are between the ages of 15 years and 18 years ; 10 per cent, between 12 and 15 ; S5 per cent, between 10 and 12, and the remainder under that age ; find the number of each class. 11. Sugar being composed of 49856 per cent, of oxygen, 43-265 per cent, of carbon, and the remainder hydrogen ; find how many pounds of each of these materials. there are in one ton of sugar. 12. At the Cambridge Borough Election, 1857, the votes given were as follows: — double votes, M. and S. 724; A. and H. 685; split votes, M. and A. 23; M. and H. 8; S. and A. 1 ; S. and H. 5 ; plumpers, M. 15 ; S. 5 ; A. 20 ; H. 4. The number Avho did not poll were 222. Find the whole number of voters on the register, and the percentage of it which each candidate obtained. 13. A merchant buys 340 loads of wheat at 8s. a bushel, 2^ per cent, of it is wasted ; he sells 56 per cent of the remainder at 7*. Gd- a bushel, 20 per cent, at 8.?. c bushel, and the rest at 10s. a bushel; what does he gain or lose by the transaction ? 14. If the increase in the number of male and female criminals be 18 per cent., while the decrease in the number of males alone is 4"6 per cent , and the increase in the number of females is 9'8. Compare the number of male and female criminals respectively. APPLICATIONS OF THE TERM AVERAGE. 261 178. Questions are often given, in which the tenn ^'Average" occurs ; a few examples of such a kind will now be worked by way of illustration, and others subjoined for practice. Ex. 1. A gentleman in each of the following years expended the following sums: in 1845 £186. 96'. 6d., in 184G £189. 0^. 7d., in 1847 £260. 15*. 4d., in 1848 £245. 4*. Gd., in 1849 £368. 5s. 6d., in 1850 £304. Is. 2d,, in 1851 £252. 6s. lid. Find his yearly average expenditure. The object is to find that fixed sum which he might have spent in each of the 7 years, so that his total expenditure in that case might be the same as his total expenditure was in the above question. Adding the various sums together we obtain the total expenditure which equals £1806. 3*. 6d. ; this sum divided by 7 gives £258. 0;?. 6d. as the average yearly expenditure. Ex. 2. In a school of 27 boys, 1 of the boys is of the age of 17 years, 2 others of 16, 4 others of 15^, 1 of 14|, 2 of 14^, 5 of 13|, 10 of 121 and 2 of 10; find the average age of the boys. The object is to find, what must be the age of each boy supposing all to be of the same age, that the sum of their ages may = the sum of the ages in the question. surn of ages in question - 17 + 32 + 62 + 14| + 29 + 68| + 122i + 20 = 366 ; .*. average age = -^ = 13f years. Ex. 3. In a class of 25 children, 19 have attended during the week. Days attended by children : 5 for 5 days, 6 for 4.^, 3 for 4, 2 for 3^, 1 for 3, 1 for 2, 1 for ^ day. Find the average number of days attended by each child. The whole number of days attended by class = (5x5 + 6x4^ + 3x4 + 2x3^ + 1x3^1x2 + 1x1) = 25 + 27+12 + 7 + 3 + 2 + 1 = 761 days; .. J 76.1 153 306 _^^, ,*. average attendance =~^ = -^77 = 77^ = 3*06 days. z5 50 100 Ex. 4. In a school the numbers for the week were : — Monday morning 67, Tuesday morn. 60, Wednesday mom. 65, Thursday morn. 68, Friday morn. 62, Monday afternoon '5 more than the average of Monday and Tuesday mornings, Tuesday aft. 59, Wednesday aft, '5 less than the 262 AHITHMETIO. average of Tuesday, Thursday the average of Monday morn, and Tuesday aft., Friday aft. 60. Find the average attendance for the week. Number of children who attended on Monday -G7 + G4; Tuesday := 60 + 59 ; Wednesday =65 + 69; Thursday ==68 + 63; Friday =62 + 60; .\ the total number of children who attended on the 10 occasions = 627 ; .*. average attendance = -— = 62'7. Ex. 5. A farm of 500 acres is let at a corn-rent equally apportioned between wheat and barley ; it is valued at £930 a year when the average price of wheat is 6s. a bushel, and that of barley 4.?. a bushel ; find tlie rent when wheat rises to the average price of 7s. 6d. per bushel, and barley to that of 5s. Sd. per bushel. First we must find the number of bushels of wheat and barley at the given rent of £930. £930 —^ — £465 the sum to be raised by each kind of grain : 465 X 20 = 155x10=1 550 bushels of wheat ; 6 465 X 20 465 X 5 = 2325 bushels of barley ; 4 .'. rent in latter case = (1550 x 7h + 2325 x 5}}s. = £1191. Us. 3d. Ex. 6. A person's average annual income from 1830 to 1850 was £374. 9*. 8d. In 1830 his income was £369. 18*. lOd., and in 1851 his income was £360. Is. Id.^ what was his average annual income from 1831 to 1851 (inclusive) ? His total income from 1831 to 1851 inclusive = £374. 9*. Sd. X 21 + £360. 1*. Id -£369. 18?. lOd. = £7854:. 5s. Sd. .*. his average mcome = ^ = £374. 0*. 3d. APPLICATIONS OF THE TERM AVERAGE. 263 Ex. LXVI. 1. In 1845 the rental of an estate amounted to £18697. 11*. 9c?., in 1846 to £'17292. 2*. Wd., in 1847 to £20185. 12*. lOd., in 1848 to £20078. 19*. 7d., in 1849 to £18582. 12*. 11<^., in 1850 to £24048.^ 5*. Id., in 1851 to £21631. 0*. Id. ; find the average rental of the 7 years. 2. The numher of quarters of grain impoi*ted into a country in 11 successive years were 2679488, 2958272, 8080298, 3474302, 2243151, 2327782, 2855525, 2588234, 8206482, 2801204, 8251901 ; find the average importation during that period. 3. If 50 quarters of wheat are sold for 77^. Sd. per quarter and 100 quarters for 78*. 2d. per quarter ; what is the average price per bushel ? 4. In a class of 23 children, 8 are boys, 15 girls. The age of the boys — 4 of 8, 2 of 11, 2 of 12. Of the girls — 5 the average age of the boys, 4 of 9, 2 of 10, 4 of 13. Find the average age of (a) the boys, (b) the girls, (d) the whole class. 5. There are 25 children on the register of one class in a school. 19 have been present at one time or other during the week. The sum of days on which the children have attended is 84^. ^Vhat is the average number of days per week attended by each child ever present during the week, there being no school on Saturday or Sunday 1 Give the answer in decimals. G. In a school of 7 classes, the average number of days attended by each child in Class I. is 4*5 ; Class II., 4 ; Class III., 3-9 ; Class IV., 4-1, • Class v., 3-6 ; Class VI., 4*2 ; Class VII., 3-3. Find the average number of days attended by each child in the school. 7. A Farm is valued at the yearly rental of £377. 10*. ; one-third of the rent is payable in money, one-fourth in wheat, and the rest in barley, the average prices being as follows : wheat Gs. a bushel, and barley 4*. Gd. a bushel. AVhat will the rent amount to when the average prices of wheat and barley are 7^. Qd. and 5s. Sd. per bushel respectively? 8. A tithe-rent of £810 per annum is commuted in equal parts into a corn-rent consisting of wheat at 56*. per qr., barley at 32*. per qr., and oats at 22*. per qr. ; find its value when wheat is at 64*. per qr., barley at -i-is. per qr., and oats at 24*. per qr. 264 ARITHMETIC. EXCHANGE. 179. Def. Exchange is the Rule by which we find how much money of one country is equivalent to a given sum of another country, according to a given course of Exchange. Def. By the Course of Exchange is meant the variable sum of the money of any place which is given in exchange for a fixed sum of money of another place : thus, for instance, in London one pound sterling, a fixed sum, is given for a variable number of French francs, more or less, ac- cording to circumstances. By the Par of Exchange is meant the intrinsic value of the coin of one country as compared with a given fixed sum of money of another. Exchanges between merchants are effected by written instruments, called Bills of Exchange ; and a bill on London entitles the holder to obtain gold in London for the value of the amount mentioned in the bill. Examples in Exchange worked out. Ex. 1. A merchant in Paris draws a bill of 1500 francs upon a mer- chant in London for goods supplied : what sterling money will the latter have to pay, exchange being 24'2o francs for £1 sterling ? Here 2-i-25 francs : 1500 francs :: £1 : required amount of money in pounds ; whence, required amount of money in pounds = £61. 17*. lyfc?. Ex. 2. What is the course of exchange between London and Lisbon when 594 milrees, 480 rces are received for £158. 16s. 9rf. ? (1 milree = 1000 rees). Here 594 mils., 480 rees : 1 mil. :: £158. I65. 2d. : course of exchange, or 594'48 mils. : 1 mil. :: S8121c?. : course of exchange, whence, course of exchange = 64 124... c?. that is, 64-] 24d or rather more than 6s. 4d. English money, would be paid for 1 milree of Portuguese. Def. Arbitration, or Comparison of Exchanges, is the method of fixing upon the rate of Exchange, called the Par of Arbitration, be- tween the first and last of a given number of places, where the course of Exchange between the first and second, the second and third, &c. of these places is known, It is called Slmple or Compound Arbitration, as three or more places are concerned. EXCHANGE. 265 Ex. 1. If the Exchange between Paris and Frankfort be at 20 francs for 9 florins, 20 kreutzers ; and the Exchange between London and Frankfort 11 florins, 54 kreutzers for the £1 sterhng, what is the course of exchange between London and Paris? (1 florin = 60 kreut- zers.) 9^ florins =^20 francs, .*. 1 florin = ^| francs, llfl. 54kreutz., orlli%fl. = £l, .-. ifl. = £^{>^, :. £J^- If francs, or £1 = (II X ^^) francs = 25*5 francs. Ex.2. £1 English being = 25*4 francs, 375 francs being =^ ]05 kreutzers, 60 kreutzers being = 1 florin ; find in English money the value of 1143 florins. 1143 florins = (1143 x 60) kreutzers, =:: (lUS X 60 X j^\ francs, = £96.85. 6K Ex. LXVII. 1. Cojivert £1519. 17s. Gd. into francs and centimes, at 23*45 francs per £. sterling. (1 franc =100 centimes.) 2. Convert 4750 milrees, 280 rees into English money, at 6-^ld. a milree. 3. Convert £246. 15*. Gd. into piastres and rials, exchange being at 472C^- a piastre. (1 piastre = 8 rials.) 4. A merchant at Lisbon draws a bill of 2000 milrees upon London. "What sterling money will the latter have to pay, exchange being 1 milree = Q8d. ? 5. If London exchanges with Holland at a gain of 6h per cent, when the course of exchange is at 35*. 6d. per £. sterling : what is the par of exchange ? 6. A bill bought in London at 25*6 francs per £. sterling, is sold in Lisbon at 172 rees per franc ; what is the exchange between London and Lisbon? 7. A merchant in London is indebted to one at St Petersburg 15,000 rubles: the exchange between St Petersburg and England is 50d. per ruble, between St Petersburg and Amsterdam 91^. per ruble, and be- tween Amsterdam and London 36*. Sd. per £. sterling: which will be the most advantageous way for the London merchant to be drawn upon ? 266 ARITHMETIC. 8. ^Vhat sum in English money must be given for 500 francs, when 25*6 francs is exchanged for £1 ? What is the arbitrated price between London and Paris, when 3 francs = 480 rees, 400 rees = 8^,9. Flemish, and 85*. Flemish = £1? 9. A person in London owes another in St Petersburg a debt of 460 rubles, which must be remitted through Paris. lie pays the requisite sum to his broker, at a time when the exchange between- London and Paris is 23 francs for £1, and between Paris and St Petersburg 2 francs for one ruble. The remittance is delayed until the rates of excliange are 24 francs for £1, and 8 francs for 2 rubles. What does the broker gain or lose by the transaction ? 10. A trader in London owes a debt of 508 pistoles to one in Cadiz : is it more advantageous to him to remit directly to Cadiz, or circuitously through France? tlie exchanges being £1=^-25-4 francs, 19 francs = 1 Spanisli pistole, 4 Spanish pistoles — £3. SQUARE ROOT. 180. The Square of a given number is the product of that number multiplied by itself Tlius 36 is the square of 0. The square of a number is frequently denoted by placing the figure 2 above the number, a little to the right. Thus G^ denotes tlie square of G, so that 6^ = 86. 181. The Square Root of a given number is a number, which, when multiplied by itself, will produce the given number. The square root of a number is sometimes denoted by placing the sign s,f before the number, or by placing the fraction h above the number, a little to the right. Thus fJ3Q or (oQ)^ denotes the square root of 3G ; so that ;^/36or (3G)^-G. 182. The number of figures in the integral part of the Square Root of any whole number may readily be known from the following con- siderations : The square root of 1 ia 1 100 is 10 10000 is 100 1000000 is 1000 ike. is ike. SQUARE ROOT. 267 Hence it follows that the square root of any number between 1 and 100 must lie between 1 and 10, that is, will have one figure in its integral part ; of any number between 100 and 10000, must lie between 10 and 100, that is, wild have two figures in its integral part ; of any number be- tween 10000 and 1000000, must lie between 100 and 1000, that is, must have three figures in its integral part ; and so on. ^Vllerefore, if a point be placed over the units' place of the number, and thence over every second figure to the left of that place, the points will shew the number of figures in the integral part of the root. Thus the square root of 99 con- sists, so far as it is integral, of one figure ; that of 198 of two figures; that of 17C432 of three figures; that of 1 70-1321 of four figures ; and so on. ISo. The following Rule maybe laid down for extracting the square root of a whole number. Rule. '' Place a point or dot over the units' place of the given num- ber, and thence over every second figure to the left of that place, thus dividing the whole number into several periods. The number of points will shev/ the number of figures in the required root (Art. 182). Find the greatest number whose square is contahied in the first period at the left ; this is the first figure in the root, which place in the form of a quotient to the right of the given number. Subtract its square from tlie first period, and to the remainder bring down the second period. Divide the number thus formed, omitting the last figure, by twice the part of the root already obtained, and annex the result to the root and also to the divisor. Then multiply the divisor, as it now stands, by the part of the root last obtained, and subtract the prodact from the number formed, as above mentioned, by the first remainder and second period. If there be more periods to be brought down, the operation must be repeated." Ex. 1. Find the square root of 1369. 1369 (37 9 67 469 469 After pointing, according to the Rule, we take the first period, or 13, and find the greatest number whose square is contained in it. Since tlie square of 3 is 9, and that of 4 is 16, it is clear that 8 is the greatest number whose square is contained in 13; therefore place 3 in the form 268 ARITHMETIC. of a quotient to the right of the given numher. Square this numher, and put down the square under the 13; subtract it from the 13, and to the remainder 4 affix the next period 69, thus forming the number 469. Take 2x3, or 6, for a divisor ; divide the 469, omitting the last figure, that is, divide the 46 by the 6, and we obtain 7. Annex the 7 to the 3 before obtained and to the divisor 6 ; then multiplying the 67 by the 7 we obtain 469, which being subtracted from the 469 before formed, leaves no remainder ; therefore 87 is the square root of 1369. Reason for the above process. Since (87)^ = 1869, and therefore 87 is the square root of 1369; we have to investigate the proper Rule by which the 37, or 30 + 7, may be obtained from the 1869. No w 1 869 = 900 + 469 = 900 + 49 + 420 = (80)2 + 7' + 2x80x7 == (30)2 + 2 X 80x7 + 7^ where we see that the 1869 is separated into parts in which the 80 and the 7, together constituting the square root, or 37, are made distinctly appai-ent. Treating then the number 1369 in the following form, viz. (30)2 + 2x80x7 + 7' we observe that the square root of the first part or of (30)^, is 30 ; which is one part of the required root. Subtract the square of the 80 from the whole quantity (80)^ + 2 x 80 x 7 + 7^ and we have 2 x 80 x 7 + 7^ remaining. Multiply the 80 before obtained by 2, and we see that the product is con- tained 7 times in the first part of the remainder, or in 2x30x7; and adding the 7 to the 2 x 30, thus making 2 x 30 + 7 or 67, this latter quantity is contained 7 times exactly in the remainder 2x80x7 + 7 or 469; so that by this division we shall gain the 7, the remaining part of the root. If we had found that the 2 x 80 + 7 or 67, when multiplied by the 7, had produced a larger number than the 469, the 7 would have been too large, and we should have had to try a smaller number, as 6, in its place. The process will be shewn as follows ; (80)2 + 2x30x7 + 72(30 + 7 (30)2 2x30 + 7 2x80x7 + 72 2x80x7 + 72 SQUARE ROOT. 2C9 This operation is clearly equivalent to the following : 900 + 420 + 49 (30 + 7 900 60 + 7 420 + 49 420 + 49 This again is equivalent to the following : 1369 (37 67 469 469 which is the mode of operation pointed out in the Rule. Note 1. The reasoning will be better understood when the Student has made some progress in Algebra. Note 2. The divisor obtained by doubling the part of the root already obtained, is often called a trial divisor, because the quotient first obtained from it by the Rule in (Art. 183), will sometimes be too large. It will be readil}'- found, in the process^ whether this is the case or not, for when, according to our Rule, w^e have annexed the quotient to the trial divisor, and multiplied the divisor as it then stands by that quotient, the result- ing number should not be greater than the number from which it ought to be subtracted. If it be, the quotient is too large, and the number next smaller should be tried in its place. N'ote 3. If at any point of the operation, the number to be divided by the trial divisor be less than it ; we then affix a cypher to the root, and also to the trial divisor, bring down the next period^ and proceed according ^to the Rule. Ex. 2. Find the square root of 74684164. {2x8 = 161 {2x86 = 172} I2x864 = l728[ 166 74684164 (8642 64 1068 996 1724 7241 6896 17282 34564 84564 Tlierefore 8642 is the square root of 74684164. 270 ARITHMETIC. Ex. 3. Find tlie square root of 71690512350825. 71690512350625 (8467025 64 |2x8 = 16[ ]2x84 = 168[ |2x846-1692[ /(2x 8467 = 16934) ] 1(2x84670 = 169840)1 164 1G86 16927 1693402 16934045 769 656 11305 10116 118912 118489 4233506 3386804 84670225 84670225 8467025 is the required square root. 184. Again, since the square root of •1 •01 •01 •0001 •COOCOl is -001 •00000001 is -OOOX &c. &e. it appears, that in extracting the square root of decimals, the decimal places must first of all be made even in number, by affixing a cypher to the right, if this be necessary ; and then if points be placed over every second figure to the right, beginning as before from the units' place of whole numbers, the number of such points will shew the number of decimal places in the root. 185. If there be no whole number, or integral part in the given num- ber, we must, in pointing, begin with the second figure from that which would be the units' place, if there were a whole number, and mark successively over every second figure to the right. If there be a whole number as well as a decimal, it will be the safest method to begin at the units' place, and point over every second figure to the right and left of it. The number of points over the wdiole numbers and decimals will shew respectively the numbers of figures in the integral and decimal parts of the root. Thus if the given number were 6115'23, place the first point over the 5, and mark from it to the right and left, thus 6115'28. If the given number were 58-432, first make the decimal places ^^ven in number thus, 58 4320, and then point thus 58-4326, SQUARE ROOT. 271 186. With the above explanation (Arts. 182 and 18^) on the subject of pointing, the rule for extracting the square root of a decimal, or of a number consisting partly of a whole number and partly of a decimal, will be the same as that before given (Art. 183) for finding the square root of a whole number. As the decimal notation is only an extension or continuance of the ordinary integral notation, and quite in agreement with it, the reason before given for the process, will in fact apply also here. 187. To extract the square root of a vulgar fraction, if the numerator and denominator of the fraction be perfect squares, we may find the square root of each separately, and the answer wdll thus be obtained as a vulgar fraction ; if not, we can first reduce the fraction to a decimal, or to a whole number and decimal, and then find the root of the resulting number. The answer will thus be obtained either as a decimal, or as a whole number and decimal, according to the case. Also a mixed number may be reduced to an improper fraction, and its root extracted in the same way. Ex. 4. Extract the square root of '4 to four places of decimals. •46060606 (-6324 86 123 400 369 1262 8100 2524 12644 57600 50576 7024 Ex. o. Extract the square root of 0006 to four places of decimals. •06O60606 (-0244 4 44 [~200 176 484 2400 1036 '404 272 ARITHMETIC. Ex. 6. Extract the square root of '0365 to five places of decimals. •0365000006 (-19104 1 ■29 1- 265 L 261 881 400 3820^ 381 t 190000 152816 87184 Ex.7. Extract the square root of 53111-8116. 53ili-8il6 (230-46 _4 43 131 129 4604 21181 18416 46086 276516 276516 Ex.8. Find the squar e root of aWi* 43 629 (23 4 129 89 240i (49 16 801 129 801 therefore square root required = ff. Ex. 9. Find the square root of f. This may be done by first reducing f to a decimal, and then by ex- tracting the square root of the decimal, thus f = '714285... •714285 (-845... 64 164 1685 742 656 I 8685 I 8425 260 or thus, SQUARE ROOT. 273 \/7 V vx7;~ 7 • 85000000 (5-916 25 109 j 1000 |_981 1181 1900 1181 11826 71900 I 70956 944 therefore , / ^ = " - ='Si5.. /5 5-91 Ex. LXVIIl. 1, Find the square roots of (1) 289 ; 576 ; 1444 ; 4096. (2) 6561 ; 21025 ; 173056. (3) 98596; 37249; 11664. (4) 998001; 978121; 824464. (5) 29506624; 14356521; 5345344. (6) 236144689 ; 282429536481 ; 282475249. (7) 295066240000; 4160580062500. 2. Find the square roots of (1) 167-9616; 28-8369 ; 57648-01. (2) '3486784401 ; 39-15380329. (3) -042849; -00139876; -00203401. (4) 5774409; 5-774409. (5) 120888-68379025; 240398*012416. 8. Extract the square roots of (1) 10; 1-6; -16; 016. (2) 235-6; '1 ; -01 , 5 ; '5. (3) -0004; -00081; 879-864. (4) 20}; 153^; }; ff^-f. (5) f;^;2^;||. (6) ^; 1^^; 23-1; 42; to four places of decimals in each case where the root does not terminate. 4. Extract the square root of '0019140625 and reduce the result to tiie corresponding equivalent fraction in its lowest terms. 5. Find the side of a square field equal in area to a rectangular field 700 yards wide and 2800 yards long„ 18 274 ARITHMETIC. 6. A square field contains lac, 22 po., 7xVyds. j find the length of its side. 7. A rectangular field measures 225 yards in length, and 120 yards in breadth ; what will be the length of a diagonal path across it ? 8. Find the length of the side of a square enclosure, the paving of which cost £27. Is. 6d. at Sd. per sq. yard. 9. The hypothenuse of a right-angled triangle is 61 yards, and the perpendicular is 24 yards, find the base. 10. A ladder, whose length is 91 feet, reaches from the extremity of a path 85 feet wide, to a point in a building on the other side, which is within 9 inches of the top of it ; find the height of the building. 11. Extract the square root of "0050722884, and find within an inch the length of a side of a square field the area of which is 2 acres. 12. Two persons start from a certain point at the same time, the one goes due east at the rate of 12 miles an hour, and the other due north at the rate of 9 miles an hour ; how far are they distant from each other at the end of six hours ? 13. A ladder 86 feet long will reach to a window 28 feet from the ground, on one side of a street ; and if the foot of the ladder be retained in the same position, will reach to a window 26 feet high on the other side. Find the breadth of the street. 14. A society collected among themselves for certain purposes a fund of £45. ISs. 9d. : each person paid as many pence as there were members in the whole society. Find the number of members. 15. The area of a circular lake is 295066 24 square yards, how many yards are contained in the side of a square of equal superficies ? CUBE EOOT. 188. The Cube of a given number is the product which arises from multiplying that number by itself, and then multiplying the result again by the same number. Thus 6 x 6 x 6 or 21 6 is the cube of 6. The cube of a number is frequently denoted by placing the figure 8 above the number, a little to the right. Thus 6^ denotes the cube of 6, 60 that 6=^ = 6x6x6 or 216. 189. The Cube Root of a given number is a number, which, when multiplied into itself, and the result again multiplied by it, will produce the given number. Thus 6 is the cube root of 216 ; for 6 x 6 x 6 is = 216. CUBE ROOT. 275 The cube root of a number is sometimes denoted by placing the sign IJ before the number, or placing the fraction ^ above the number, a little to the right. Thus ^/216 or (216)* denotes the cube root of 21G; sothat;^2T6or(216)* = 6. 190. The number of figures in the integral part of the cube root of any whole number may readily be known from the following con- siderations : The cube root of 1 is 1 1000 is 10 1000000 is 100 ICOOOOOOOO is 1000 &c. is &c. Hence it follows that the cube root of any number between 1 and 1000 must lie between 1 and 10, that is, will have one figure in its in- tegral part; of any number between 1000 and 1000000, must lie between 10 and 100, that is, will have two figures in its integral part; of any number between 1000000 and 1000000000, must lie between 100 and 1000, that is, must have three figures in its integral part; and so on. Wherefore, if a point be placed over the units' place of the number, and thence over every third figure to the left of that place, the points will shew the number of figures in the integral part of the root. Thus the cube root of 677 consists, so far as it is integral, of one figure ; that of 198999 of two figures ; that of 134198999 of three figures; and so on. 191. The following Rule may be laid down for extracting the Cube Root of a whole number. Rule. " Place a point or dot over the units' place of the given num- ber, and thence over every third figure to the left of that place, thus dividing the whole number into several periods. The number of points will shew the number of figures in the required root. (Art. 190.) Find the greatest number whose cube is contained in the first period at the left ; this is the first figure in the root, which place in the form of a quotient to the right of the given number. Subtract its cube from the first period, and to the remainder bring down the second period. Divide the number thus formed, omitting the last two figures, by 3 times the square of the part of the root already obtained, and annex the result to the root* 18—2 276 AEITHMETIC. Now calculate the value of 3 times the square of the first figure in the root (which of course has the value of so many tens) + 3 times the product of the two figures in the root + the square of the last figure in the root. Multiply the value thus found by the second figure in the root, and subtract the result from the number formed, as above mentioned, by the first remainer and the second period. If there be more periods to bo brought down the operation must be repeated." Ex. 1. Find the cube root of 15625. 15625 (25 2'^ = 8 3x2=^-12 p7625 3 X (20)2 = 3x400 -1200 3x20x5:= 800 5^= 25 1525 Multiply by 5^ 7625 7625 After pointing according to the Rule we take the first period, or 15, and find the greatest number whose cube is contained in it. Since the cube of 2 is 8, and that of 3 is 27, it is clear that 2 is the greatest number whose cube is contained in 15 ; therefore place 2 in the form of a quotient to the right of the given number. Cube 2, and put down its cube, viz. 8, under the 15 ; subtract it from the 15, and to the remainder 7 affix the next period 625, thus forming the number 7625. Take 3 x 2\ or 12, for a divisor ; divide 76 by 12, 12 is contained 6 times in 76 ; but when the other terms of the divisor are brought down, 6 would be found too great, therefore take 5. Annex the 5 to the 2 before obtained ; and calculate the value of 3 x (20)^ + 3 X 20 X 5 + 5^ which is 1525 ; multiplying 1525 by 5 we obtain 7625, which being subtracted from 7625 before formed leaves no remainder, therefore 25 is the cube root required. Reason for the above process. Since (25)^:::= 15625, and therefore 25 is the cube root of 15625; we have to investigate the proper Rule by which the 25, or 20 + 5, may be obtained from 15625. Now 15625 =^ 8000 + 7500 + 1 25 ^ 8000 + 6000 +1500+125 -.(20)3 + 3 X (20)'^ X 6 +3 X 20 X b'' + b^ CUBE ROOT, 277 where we see that the 15625 is separated into parts in which the 20 and the 5, together constituting the cube root, or 25, are made distinctly ap- parent. Treating then the number 15625 in the following fornij viz. (20)3 + 3 X (20)2 X 5 + 3 X 20 X 5^ + 5^ we observe that the cube root of the first part or of (20)"' is 20 ; which is one jxart of the required root. Subtract the cube of the 20 from the whole quantity, and Ave have 8 x (20)^ x 5 + 3 x 20 x 5- + 5^ remaining. Multi- ply the square of the 20 before obtained by 8, and we see that the pro- duct is contained 5 times in the first part of the remainder, or in 8 X (20)^ X 5; and adding 8 times the product of the two terms of the root + the square of the last term of the root, thus making 8x(20)^ + 3 X 20 X 5 + 5^ we see that this latter quantity is contained 5 times exactly in the remainder 8 x (20)^^ x 5 + 8 x 20 x 5' + 5^, so that by tliis division we shall obtain the 5, the remaining part of the root. The process will be shewn as follows : (20)3 + 3 X (20)=* X 5 + 3 X (20) X 52 + 53 (20 + 6 (20)3 divisor = 3x(20)^ ^ 3 X (20)2x5 , ""^-T^-(2oF^^^ .-. ]3x (20)^+3 X 20 X 5 + 52} x5 = 3 <(20)' x5 + 3> 20 <52 + 5^ 3 X (20)2 x5 + 3> (20 X 52 + 53 This operation is clearly equivalent to the following : 8000 + 6000 + 1 500 + 125 (20 + 5 8000 3 X (20)2 = 1200, and = 5 6000 + 1500 + 125 (1200 + 800 + 25) X 5= 6000 + 1500 + 123 This again is equivalent to the following : 15625 (25 3 X 22^=3x4-12, and if = 5 3 X (20)2 = 1200 3x20x5= 300 8 7625 + 52= 25 1525 5 7625 7625 which is the mode of operation pointed out in the Rule. 278 ARITHMETIC. Note 1 . The reasoning will be better understood when the student has made some progress in Algebra. Note 2. The divisor which is obtained according to the Rule given in (Art. 191) is sometimes called a trial divisor, because the number from the division may be too large, as was the case in the above Example, in which case we must try a smaller number. We shall readily ascertain whether the number obtained from the division is too large or not, because if it be too large, the quantity which we ought to subtract from the number formed by a remainder and a period will turn out in that case to be larger than that number, which of course it ought not to be, and so we must try a smaller number. Note 3. If at any point of the operation, the number to be divided by the trial divisor be less than it ; we affix a cypher to the root, two cyphers to the trial divisor, bring down the next period, and proceed according to the Rule. Ex. 2. Extract the cube root of 95443093. 95448993 (457 43 = 64 trial divisor 3x(40y = 4800 8x40x5= 600 52= 25 3x4^ = 48 5425 5 27125 trial divisor -.3 X (45)='=6075 Now 45 has the value of 450 ; .-. 3 X (450)2 = 607500 3x450x7= 9450 7'= 49 31443 314 48 goes 6 times, but 6 will be found too larger- try 5. 27125 616999 7 4318993 4318993 43189 -TTT^rr-goes 7 tmies, and we b075 are led to conclude that 7 is the figure, because 7^ = 343, and 3 is the final figure in the remainder. 4318993 Therefore 457 is the cube root required. CUBE ROOT. 279 Ex. 3. Find the cube root of 223648543. 228648543 (607 6^ = 216 rial divisor = 3 x 6^ rial divisor =^3 X (60)2 - 3 X (600)2 = 1080000 3x600x7= 12600 7'= 49 = 108 -10800 1092649 7 7648543 7648 7648543 7648543 70 is not divisible by 108 ; bring down the next period and affix to the root ; iM-ou goes 7 times, and 7 seems likely to be the figure required ; since 7^ = 343, and 3 is the final figure in the remainder. Therefore 607 is the cube root required. 192. Again, since the cube root of '001 is '1, the cube root of '000001 is -01, the cube root of '000000001 is '001, &c. is &c. it appears, that in extracting the cube root of decimals, the decimal places must first of all be made three, or some multiple of three in number, by affixing cyphers to the right, if this be necessary ; and then if points be placed over every thii-d figure to the right, beginning a.s before from the units' place of whole numbers, the number of such points will shew the number of decimal places in the cube root. 193. If there be no whole number or integral part in the given num- ber, we must in pointing begin with the third figure from that w^hich would be the units' place, if there were a whole number, and mark successively every third figure to the right. If there be a whole number as well as a decimal, it wall be the safest method to begin at the units' place, and point over every third figure to the right and left of it : tlie number of points over the whole numbers and decimals will shew respectively the numbers of figures in the integral and decimal parts of the root. Thus if the given number were 5623'453134, place the first point over the 3, and mark from it to the right and left, thus 5623*453134. If the given number were 5*23, make the number of decimal places equal to 8, by affixing a cypher thus, 5'230 ; place the first point over the 5, and the second over the : if the root to more decimals than one is required, more cyphers must be affixed. 280 ARITPIMETIC. 194. With the above explanation (Arts. 190, 192) on the subject of pointing, the rule for extracting the cube root of a decimal, or of a number consisting partly of a whole number and partly of a decimal, will be the same as that before given (Art. 191) fcr finding the cube root of a whole number. As the decimal notation is only an extension or continuance of the ordinary integral notation, and quite in agreement with it, the reason before given for the process, will in fact apply also here. 195. To extract the cube root of a vulgar fraction, if the numerator and denominator of the fraction be perfect cubes we may find the cube root of each separately; and the answer will thus be obtained as a vulgar fraction ; if not, we can first reduce the fraction to a decimal, or to a whole number and decimal, and then find the root of the resulting number. The answer will thus be obtained either as a decimal, or as a whole number and decimal, according to the case. Also a mixed number may be reduced to an improper fraction, and its root extracted in the same way. Ex. 4. Find the cube root of 48228-544. 48228-544 (36-4 33 = 27 3x3^-27 21228 3 X (30)^ -2700 8x30x6= 540 6^= 86 8276 6 19656 - 19656 3 X (36)2 = 8880 1572544 8 X (360)2 = 380800 3 X 300 X 4 = 4320 42= 16 393136 1 4 1572544 1572544 Therefore 36 "4 is the cube root required. CUBE ROOT. 281 Ex. 5. Find the cuije root of 000007 to three places of decimals. •000007000 (-019 3xr=.3 Gx(10)2- 300 3x10x9:^ 270 9^= _8]. 651 •^ 5859 GOOD 6859 141 Ex. 6. Find the cube root of ^ to three places of decimals. /^ -555555505. .. •555555555 (-822 83 = 512 3x8^ = 192 3 X (80)2 = 19200 3 X 80 X 2 = 480 2'= 4 19684 2 39368 8 X (82)2 = 20172 3 X (820)- = 2017200 3 X 820 X 2 = 4920 2^ = 4 2022124 _2 4014248 43555 39368 4187555 4044248 143807 196. Higher roots than the square and cube can sometimes be extracted by means of the Rules for square and cube root ; thus the 4tli root is found by taking the square root of the square root ; the 6th root by taking the square root of the cube rootj and so oij. 282 ARITHMETIC, Ex. LXIX. 1. Find the cube roots of (1) 1728; 3375; 29791. (2) 54872; 110592; 300763. (3) 681472; 804357; 941192. (4) 2406104; 69426531 ; 8365427 (5) 251239591; 28372625; 48228544. (6) 17173512; 259694072; 926859375. (7) 27054036008; 219365327791. 2. Find the cube roots of •389017; 52-461759; 95443-993; -000912673; •001906624; '000024389. 3. Find the cube roots of (1) 3, -3. -03. (2) |; |«;44U (3) 405n^;7i; 3-00415. (4) -OOOljilUg?, to three places of decimals, in those cases where the root does not terminate. 4. Find the cube root of 233744896, and also the cube root of the last-mentioned number multiplied by 008. 5. The cost of a cubic mass of metal is £10481. 1^. M. at 10*. 5d. a cubic inch. What are the dimensions of the mass ? 6. A cubical block of stone contains 50653 solid feet, what is the area of its side ? 7. A cube contains 56 solid feet, 568 solid inches ; find its edge. 8. Find the cost of carpeting a cubical room, whose content is 21717639 solid feet, with carpet 21 inches broad, at 3*. 6d. a yard. 9. A cubical box contains 941192 solid inches : find the cost of painting its outside surface at 6d. a square foot. 10. If the solid content of a cube be 37 ft. 64 in., shew that its surface will be 66 ft. 96 in. 11. The edge of a cubical vessel is 2 feet long: what is the length of the edge of another cubical vessel containing 3 times as much ? 12. Find the 4th root of 43046721 ; and the 6th root of '000000004096. MISCELLANEOUS QUESTIONS AND EXAMPLES. 283 Ex. LXX. Miscellaneous Questions and Examples on preceding Arts. I. 1. Explain how compound subtraction would be facilitated by the introduction of a decimal coinage. Subtract 5 florins, 5 cents, 5 mils from 9 florins, 6 cents, and shew that 8 times the diff'erence equals £S. 8s. 2. What is the whole value of 6 J yds. of cloth at 18.?. Gd. a yard, 10| lbs. of tea at 5*. 4c?. a lb., and 5 qrs. 8 bush, of corn at dQs. a quarter ? Divide the sum among 4 people in the proportions 1, 2, 8, 4. 3. Assuming only the definition of a vulgar fraction, prove that the numerator and denominator of any vulgar fraction may be multiplied or divided by the same integer without altering its value. (a) What fraction of a sovereign is 4i^ — 10|4 + 9^ — j\2^ of a penny ? (^) Find the value of J x U of || + (21 + ^) x ^. 4. The profits of a tradesman average £28. Ss. 2d. per week ; out of which he pays 5 persons at the rate of 1 guinea, and 8 others at the rate of 17*. Qd. per week respectively ; his yearly outgoings for rent, &c. amount to £861. 11*. lOd. Find his net annual income. 5. If 10 men or lo boys can reap 20 acres of corn in 6 days working 14 hours a day, how many boys must be employed to assist 3 men to reap 6 acres in If days of 8 hours a day ? 6. What is the height of a closet 8I- ft. by 6| ft. which will exactly contain 12 boxes 4^ ft. long, 8^ ft. wide, 2i ft. deep ? 7. Two lines are 41*06328 and •04G8 of an inch long respectively. How many lines as long as the latter can be cut off from the former ? What will be the length of the remaining line ? 8. Explain the method of extracting the cube root of a number. Find the area of the surface of a cube which contains 733626753859 cubic inches. 9. Shares in a certain Railway pay £3. 5s. dividend per annum. How much must I give for them to get 5 per cent, for my money ? A person having bought 20 shares at this price sells them when they have risen £7 each, and buys 8^ per cent, stock at 90. Find the change in his income. 284 ARITHMETIC. 10. What sum of money will amount to £845 in 2 years at 4 per cent, compound intci'est, and what will it amount to in 2 more years? 11. A merchant sells 72 quarters of corn at a profit of 8 per cent., and 37 quarters at a profit of 12 per cent. ; if he had sold the v/hole at a uniform profit of 10 per cent, lie v/ould have received £2. 14s. 8d. more than he actually did ; what was the price he paid for the corn ? 12. The gross receipts of a railway company in a certain year are apportioned as follows ; 41 per cent, to pay the working expences, 56 per cent, to give the shareholders a dividend at the rate of Si- per cent, on their shares ; and the remainder, £15000, is reserved ; find the paid-up capital of the company. II. 1. Express in figures one billion, three hundred thousand millions, five hundred and seven thousands, three hundred and sixty four ; and in writing 236045978218478. 2. "V^^lien the pound sterling was worth 24 francs, 75 centimes, a traveller at Dover received 15*. for a Napoleon (20 francs). Of how much was he cheated ? 8. Shew how by first principles to calculate values by Practice. Find by Practice the value of 750 articles at £5. Ss. M. each ; and the price of 3 cwt. 2 qrs. 18Hbs. at £3. 7*. Qd. per cwt. 4. Explain the diff'erence between a Vulgar and a Decimal Fraction. Simplify 2-4=- 2-'^ rJJ. _i- ' J_ 2-6 ■ 8-7* (/3) (5) -0576 X 1-97 + -142857 - 2i + -0454864. If the latter result represent a square in yards, find the length of its side in inches. 5. A and B can finish a piece of work in 11 days, A and C in 2 daj's, and B and C in 8 days. If Qs. be paid for the piece of work, what are a day's wages of each workman ? 6. A tax of £580 is to be raised from 8 towns, the numbers of inhabitants of which are respectively 2500, 8000, and 4200. How much should each town pay, and each person in it ? MISCELLANEOUS QUESTIONS AND EXAMPLES. 285 7. If 15 men or 40 boys do a piece of work iii 12 days, how many days would 10 men and 20 boys take to do a piece of work 7 times as great ? 8. Define Interest and Discount. Shew that the Interest and Dis- count on £64:. Ws. for 8 months at 4^ per cent, per annum, differ by Is. lid. nearly. 9. The breadth of a room is 14 ft. ; the cost of papering the walls at 1^. a square yard is £4 ; and that of carpeting the room at 4s. 6:/. a square yard is £5. 12*. Determine the height and length of the room. 10. Explain the following extract from the ' Times ' of January 8, 1857: "Consols which left off last evening at 94|- to } opened at 04 to I, and remained without variation to the close of business." A person has 200 shares in the North Devon Railway for which he gives £100 per share. When they are paying £2 per cent, he sells them all at £46 per share, and invests the proceeds in the 3 per cent, consols at 92. Find the alteration in his income. 11. A fixed rent of £1170 per annum is converted into a corn-rent of one half wheat at the average price of 48*. per quarter, and the other half barley at the average price of 30*. per quarter ; what will be the rent when wheat has advanced to oijs. and barley to 32*. per quarter ? 12. If the estimated annual value of the property in a certain parish consist of the yearly rent paid to the landlord together with the rates, and the rates be calculated upon the rent after a reduction of SO per cent ; find the rateable value of a tithe-rent charge, the estimated annual value of which is £884 per annum, when the rates amount to 3*. in the pound. III. 1. Shew from first principles how to divide one fraction by another. Prove that the fraction ;z — - is greater than f and less than |-. Simplify 11 + T2 ' "^ "^'14x3 15 itA^.^fJL^A 2. Express («) (^ + f )£ + (I + i>' + {\ + i)d. as the decimal of £1. (/3) 60 francs as the decimal of a guinea, £1 being equivalent to 25 francs. 286 ARITHMETIC. 8. A man contracts to perform a piece of work in 60 days, and immediately employs upon it 80 men ; at the end of 48 days the work is only half done ; required the additional number of men necessary to fulfil the contract. 4. The price of posting in Germany being 1^ florins per German mile, which =4.} English miles; find the cost in English" money of posting 881 English miles in Germany. £1 English =^25-4 French francs; 8"75 francs =105 kreutzers; 60 kreutzers — 1 florin. 5. A can do a piece of work in 12 hours, 5 in 4 hours, and C in 3 hours. A, B and C all work togetlier for half an hour, when A leaves off. How long will it take B and C to finish the piece of work ? 6. Explain the method of pointing in extracting the square root of a whole number, and also of a decimal. (a) The surface of a cube is 86*64 square feet, find the length of its edge. (/9) Given that the square of 10129 is 102596641, find the square of 101293 without going through the operation of squaring. (7) Given that the square root of 105625 is 825, find that of 10582009. 7. Define Pi-esent Worth. A person invests the Present Worth of £30192 (due 6 months hence at 4 per cent, per annum) in the 3 per cent. Consols at 92^. What will be his half yearly dividends after the deduction of an Income Tax of Is. M. in the £. ? 8. If a cubic foot of marble weigh 2716 times as much as a cubic foot of water, find the weight of a block of marble 9 ft. 6 in. long, 2 ft. Sin. broad, 2 ft. thick, sui)posing a cubic foot of water to weigh 1000 oz. 9. A bankrupt has book-debts equal in amount to his liabilities, but on £6000 of them he can only recover 18*. 4af. in the pound, and the expences of the bankruptcy are 5 per cent, on the book-debts ; if he pay 13s. in the pound what is the amount of his liabilities ? 10. A publisher wishes to net 14*. for each copy of a work ; what price should he put upon it so as to be able to allow the trade 30 per cent, discount ? 11. A man, buying goods, by means of false scales defrauds to tho extent of 15 per cent., and 15 per cent, in selling; find his whole gain per cent. MISCELLANEOUS QUESTIONS AND EXAMPLES. 287 12. Which would be the better investment, 8 per cent, stock at 87 subject to an Income Tax of IGd. in the pound, or railway shares at £230 each, yielding annually £7. 10-y. clear of Income Tax ? IV. 1. A stationer bought 40 reams of paper at 12*. 6d. a ream, and 60 reams at 15*. 6d. a ream ; find the whole cost, and the average price per ream, and if the whole be sold at 15*. a ream, find the profit. 2. The following questions are to be worked decimally, and the answers given in the decimpi coinage :— (a) A bankrupt's effects were worth £4265, and his estate paid three dividends of 2fl. 5 c. ; Ifl. 1 c. 8 m.; and 2fl. 9 m. in the pound respectively ; what was the whole loss sustained by his creditors ? (/3) If £2843. 7 fl. 5 c. be due from London to Paris when £1 is worth 25 francs, how much must be remitted when a guinea is worth 27 francs ? 3. Three men, working 9 hours a day, take 16 days to pave a road 315yds. long and 80ft. broad; how many days will four men, two of whom work 8 hours, and two 10 hours a day, take to pave a road 1575 yds. long, and 35 ft. 6 in. broad ? 4. The areas of two cubes are respectively 5359*375 and 5 •359375 cubic feet ; find the difference of the lengths of their edges in inches. 5. A person bought 4 Railway tickets to go 60 miles. Two were for the 1st Class, one for the 2nd, and the fourth a half first class ticket for a child. The cost of a second class ticket was § of that of a first class, and the whole sum was £1. 11*. 8d. Find the price of each ticket, and the rate per mile for the 1st Class. 6. When are four quantities said to be in proportion ? Shew by means of your definition that £191. 12*. 6d. : £31. 10*. :: 365 days : 60 days ; and deduce the method of working the following question : If 3 workmen earn between them £191. 12*. 6d. in a year, in what time will they earn £31. lOi.? 7. The Discount on a sum due one year hence at 5 per cent, per annum interest is £15. What is the sum ? 8. If 8 variegated silk scarfs, measuring each 3 cubits in breadth and 8 in length cost 100 nishcas ; what will a like scarf 8^ cubits long and AKITHMETIC. 1 a cubit wide cost in terms of drammas^ panas, cacinis, and cowry- shells ? 1 nishca = 16 drammas, 1 dramma = 16 panas, 1 pana = 4 cacinis^ 1 cacini = 20 cowry shells. 9. A person invests a sum of money in 50 casks of sugar each con- taining 11 cwt. Sqrs. 2 lbs. at 17^. ll^d. per cwt., what price must he sell them at after 6 months to realize the same interest as he might have had for his money at 4i- per cent.? 10. It is agreed that the rent of a farm shall consist of a fixed sum together with the value of a certain number of bushels of wheat ; when wheat is 56^. a quarter the rent is £250, when wheat is GOs. a quarter the rent is £260, what will the rent be when wheat is 80^. a quarter 1 11. A and B can do a piece of work in 10 days ; B and C in 15 days and A and C in 25 days ; they all work at it for 4 days ; A then leaves, and B and C go on for 5 days ; B then leaves : In how many days will C finish the work ? 12. A ship's hold is 99 ft. long, 40 ft. broad, and 5 ft. deep, how many bales can be stowed in it each S ft. 6 in. long, 2 ft. 8 in. broad, and 2 ft. 6 in. deep, leaving a gangway of 4 ft. broad ? V. 1. (a) The French metre being 89*37 in., how many yards are there in 8600 metres ? (/3) 3 versts being = 2 mileS) in w^hat time will a man travel over 2500 versts at the rate of 10 miles an hour ? 2. State what fractions produce terminating decimals, and what produce recurring decimals. Explain the reason. Reduce to decimals the vulgar fractions f , f^, f^, and add them; and divide their sum by '00003741 to two decimal places. 3. A silversmith purchases a large dish w^eigliing 80 oz., and forms it into 2 dozen of dessei-t-spoons, and one dozen of table-spoons. If the latter weigh 28 oz., what is the weight of each dessert-spoon, and what is its value at ^^ of a penny per grain ? 4. Add together f of f of £2. 5s., ^ of 3 guineas, '27 of £l. 18?. 6(/., and 2-154 of £2. 15*., and reduce the result to the decimal of 25 guineas. 5. How much may a pei-son who has an annual income of £840. 5fl. spend per day, in order to save £63. 9 fl., 6 c, 6| m, after paying an In- come Tax of 16d. in the £. ? MISCELLANEOUS QUESTIONS AND EXAMPLES. 289 6. Find the square root of — ^Tu)?) — ' ^ "^^ ^^"^ ^^°^ ^^ '^^' 7* If a piece of work can be finished in 45 days by 35 men, and if the men drop off by 7 at a time at the end of every 15 daySj how long will it be before tlie work is finished ? 8. Divide £16984 among A, B, C and D ; so that A's, share : -B's share :: Q : b ; B's share : C's share :: 2 : 3 ; and C's share : D's share :: 4 : 3. 9. What is the cost of paper for the walls of a room 30 ft. long, 15 ft. broad, and 15 ft. high, the paper being li^yds. wide, and its price 4|d per yard? What would be the cost for a room twice as long, twice as broad, and twice as high, the paper twice as wide, and costing twice as much per yard as before ? 10. If when 25 per cent, is lost in grinding wheat, a country has to import ten million quarters, but can maintain itself on its own produce if only 5 per cent, be lost, find the quantity of wheat grown in the country. 11. How many flag-stones, each 5*76 ft. long and 4'15 ft. wide are required for paving a cloister which encloses a rectangular court 4577 yds. long and 41 '93 yds. wide ; the cloister being 1245 ft. wide? 12. (a) A man wishing to invest «£1000 in the 3 per cent, consols inquires the price of the stock, and finds it to be 86 per cent, ; he delays the investment however until the consols have risen to 87. What efifect has the delay on his income ? (/5) The value of money increases from 4 to 5 per cent.; supposing this to have a corresponding influence on the funds, how much ought the 8 per cent, consols to sink ? VI. 1. Explain our decimal system of arithmetic, and how it is that we are enabled with digits to express any number, however great. 2. If 12 men or 18 boys can do f of a piece of work in 6^ hours, in what time will 11 men and 9 boys do the rest? 3. If Napoleons can be bought in London at 16*. 6d. each, and 5 thalers 17| groschen can be obtained for each in Berlin, where the sove- reign is worth 6 thalers 20 groschen, what sum would be gained upon each Napoleon by the operation ? (1 thaler = 30 groschen.) 4. The net rental of an estate, after deducting 7c?. in the pound for Income tax, and 5 per cent, on the remainder for the expenses of collect- ing, is £959. 3*. 8d., find the gross rental. 19 ARITHMETIC. 5. Define discount. If the discount on £226. 2^. Sd. due at the end of a year and a half be £12. 16*., what is the rate of interest ? 6. A has stock in the 3 per cent, consols which produces him £300 annum. He sells out one half at 92, and invests the proceeds in the South Devon Railway when a £50 share is worth £23. What dividend per cent, per annum ought the South Devon Railway to pay so that he may increase his income £50 per annum by the operation ? 7. A grain of pure gold can be drawn out into a wire 550 feet long; find the cost of a wire of the same thickness which would extend round the earth, assuming the circumference of the earth to be 25,000 miles. and the value of gold to be £4. 5s. per oz. troy. 8. (a) If ^ = ]!- of B, and C - 2^ of B, find the ratio of A to C. (p) Simplify j-.^^L^|_|^±).4f. (7) Divide 10-886 by 51-6 and 1083-6 by 5*16 and also by -00516, and prove each result by vulgar fractions. 9. A shopkeeper buys I; cwt. ot tea at 4*. 2d. per lb., and mixes it with tea which cost him 2*. lie?, per lb. How much of tlie latter must he add to the former that he may sell the mixture at 3s. 8d. per lb., and gain 20 per cent, on his outlay ? 10. 4 ft. 4 in. being the area of a map which is laid down on the Scale of an inch to a mile, required the number of acres represented. 11. (a) What must be the market value of the 3 per cent, consols in order that after deducting an Income Tax of 1*. Ad. in the pound, they may yield 4 per cent, interest? (/3) After paying an Income Tax of 10 per cent, a person has £1250 a year, find his gross income. 12. In an election of a member of parliament yV ^h of the constituency refused to vote, and of two candidates the one who is supported by i§ th of the whole constituency is returned by a majority of five, find the number of votes for each. VII. 1. Prove that 29 multiplied by 15 - 15 multiplied by 29. What h the difference between abstract and concrete numbers ? 2. On the roof of Covent Garden Theatre there was a tank holding 18 tons of water. Supposing it cubical what would have been its dimen- sions 1 One cubic foot of water weighs 1000 oz. MISCELLANEOUS QUESTIONS AND EXAMPLES. 291 3. If it take 3' to read over two pages of a book containing 30 lines in each page, witli an average of 10 words in a line, how many pages of another book can be read in 20' when there are 50 lines in a page and 12 words in a line ? 4. Required the expense of painting the outside of a cubical box, whose edge is 8'5 ft., at I'S^. per sq. yd. 5. The wages of 25 men amount to £76. 18.?. 4c?. in IG days, how many boys must work 24 days to receive £103. 10*., the daily wages of the latter being one half those of the former ? 6. If a bookseller gain fth of the prime cost of a book by selling it at 0*. Gd.j what would be his gain per cent, if he sold it at 6s. QdA 7. Brussels carpet is 2Ht. wide, and costs 5s. per yd.; Kidderminste> carpet is 8 ft. wide, and costs 3*. 4^d. per yd. ; drugget is 4 ft. wide, and costs 2*. 6d. per yd. These carpetings will last 10 yrs., 6 yrs., and 8 yrs. respectively ; which is the cheapest and which the dearest in wear in the long run? 8. (a) Simplify ; ^ | -^ [ ^ ^ , and reduce to its lowest terms .2^-1 of 1| _ 1\ J_ \iof3^ + M W H' (/3) Find the value of i of 16*. 6ld. + ^ of 12*. IQid + J of £2. 4*. 8|rf; and of (#0 of 11-8-/,^ of ir02)-0 1. 9. (a) A person investing in the 4 per cents, receives 5 per cent, for his money ; what is the price of stock ? (J3) "When the 3 per cents, are at 80, how much stock must be sold out to pay a bill of £690. 3*. M. due 9 months hence at 3 per cent, simple interest? 10. A merchant has teas worth 5*. and 8*. 6d. per lb. respectivelj% which he mixes in the proportion of 2 lbs. of the latter to 1 lb. of the former. How much will he gain or lose per cent, by selling the mixture at 4*. 6d. per lb. ? 11. A and B set out from the same place in the same direction, A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days ; he then turns again, and pursuing his journey overtakes B at the end of 22.^ days after the time they first set out. Shew that B uniformly travelled 10 miles a day. 19—2 292 ARITHMETIC. 12. A, B and C do f th of a piece of work together in 24 days, A does the same amount of Work as B does in the same time, had either ^ or ^ been absent, then the two others would have accomplished f th of the work in 28 days. In what time can each separately do the work ? VIII. 1. What is meant by reducing one quantity to the fraction of another ? (a) What fraction is 1*. 6|d of 2s. M. ? and 5^ of 41? and ii of 4.ld. of ds. 1U. 1 and i\ of £62. 1.9. 7M of £5 ? (^) If A be 2| of B, and 5 be" 1| of C, and D be 7^ of C, what fraction is ^ of Z) ? (7) If 2| of ^= 1 J of (^ + 1 of A), find two whole numbers which shall bear to each other the ratio of A to B. 2. A pound of silver is coined into QQ shillings, of which 62 only are issued. If 19 half-crowns, and 15 sixpences are melted into bullion, and sent to the Mint to be recoined, what sum will be re-issued ? 3. A person rows a distance of 1|^ miles down a stream in 20 minutes, but without the aid of the stream it would have taken him half an hour; what is the rate of the stream per hour? and how long would it take him to return against it ? 4. The shares in a speculation are £3. l^s. A person buys 77 shares when they are at 4 per cent, below par, and sells them at 1 per cent, premium, what is his gain ? 5. A and B engage to do a piece of work for 30*. A could do the work alone in 4 days, and ^ in 5 days ; with the help of a boy it is com- pleted in 2 days ; how should the money be divided ? G. A bill of £999 is due in such a time that £80 would in the same time amount to £83. 5*. What discount should be allowed for ready payment ? 7. If a clergyman commute his tithes, valued at £500, for wheat, barley, and oats in equal portions, what quantity of each grain will he receive, supposing the averaLje price of wheat to be 6s. 6d., barley 3*. 9d.y and of oats 2*. 9d. a bushel ? In the above question what will be the value of his living when the price of each grain is advanced 1*. per bushel ? 8. A room 24 ft. 7 in. long, 20 ft. 5 in. broad, 15 ft. high, is to be papered ; there is a door in it 6 ft 6 in. by 3 ft., and 3 windows, each MISCELLANEOUS QUESTIONS AND EXAMPLES. 293 1 1 ft. 9 in. by 2 ft. 10 in. Required the cost of papering the room at 2*. 4^d. per sq. yard. 9. How many times does -0009 of a shilling exceed -0000003 of a shilling? 'What number will represent 116"0185 grains when 4'001.5 grains is the unit of weight ? Determine the heaviest unit for which •006i ounces will be represented by an integer. 10. (a) Extract the square roots of 16-016004 ; (2) -027. (/3) Extract the cube roots 512-768384064 ; (2) 42|. (7) The edges of a rectangular chest which contains 64 cubic ft., are in the proportion of 1, 2, 4 ; find the actual length of its edges. 11. A ship 40 miles from the shore springs a leak which admits 8| tons of water in 12 minutes. 60 tons would suffice to sink her, but the ship's pumps can throw out 12 tons of water in an hour. Find the average rate of sailing so that she may reach the shore just as she begins to sink. 12. ^ in 2 days can do as much as C in 3 days, and 5 in 5 days as- much as C in 4 days ; what time would B require to finish a piece of work which A can do in 12 weeks ? IX. 1. Write down a rule for working examples — 1st, in Simple Fellow- ship ; 2nd, in Compound Fellowship. (a) A ship worth £1800 being entirely lost, of which ith belonged to A, Jth to B, and the rest to C; find the loss which each will sustain if she be insured for £1080. (iS) A and B each invest a certain sum of money in a business. The sum which A invests is f of that which B invests. At the end of 7 months A withdraws } of his capital, and at the end of 9 months B withdraws } of his. The profits at the end of the year are £132. 12*.; how ought they to be divided ? 2. A person buys 3 lbs. of tea at 4^. 5d. per lb., and mixes them with 5 lbs. of tea at 2*. 10c?. What will 2 lbs. of his tea cost him ? 3. A person contracts to make a railway 189 miles long in 15 months. He employs 129 men, but after 3 months finds that he has only finished 28 miles. How many men must he employ to finish it within the time required ? 4. A pound troy of English standard gold, ^Mhs fine, is worth £46. 12s. 6d., find the value of a coin weighing 7 dwts. 11 grs. in which the per centage of fine gold is 92 "^ 294 ARITHMETIC, 6, A cistern lias 3 pipes, A^ B, and C ; A and S can fill it in 3 aiid 4 hours respectively; and C can empty it in 1 hour. If these pipes be opened in order at 3, 4, and 5 o'clock, when will the cistern be empty? 6. How many parcels of 6 lbs. and 8 lbs. each can a grocer make out of a hogshead of sugar weighing 4 cwt. 3 qrs. 14 lbs., so as to have the same number of parcels of each sort ? 7. (a) If the interest on £264 for 20 days be 10^. 9d. what is the rate per cent, per annum? (/3) In how many years will £936. 1 3^. 4c?. amount to £11 67 7^. Ud. at 4f per cent, per annum ? (7) ^Fhat must be the rate of interest in order that the discount on £387. 7^- 7ld. payable at the end of 3 years may be £41. 10*. l^d.l 8. Of 138,918 persons, 30'66 per cent, can read and write; 68-89 per cent, can do neither ; and the rest can only read ; find the numbers in each class. 9. If gold be beaten out so thin that an oz. avoird. will form a leaf of 20 sq. yds., how many of these leaves will make an inch thick, the weight of a cubic foot of gold being 10 cwt. 95 lbs. ? 10. A person bought goods on the continent; the cost of freight and insurance was 15 per cent., and that of duty 10 per cent. On the original outlay; he was obliged to sell them at a loss of 5 per cent.; but if he had made £3 more of them he would have gained 1 per cent. What was the original outlay? 11. (a) If 60 guns firing five rounds in 8 minutes kill 350 men in IJ hours, how many guns firing 7 rounds in 9 minutes will kill 980 men in 25 minutes at the same rate 1 (/3) If the Income Tax be 7d. m the pound in the first half of the year, and S^d. in the second, what is the net income of a gentleman whose gross annual receipts are £1542. 10*. 6d. 1 12. The expense of constructing a railway is £2,000,000, of which ^th part was borrowed on mortgage at 5 per cent, and the remaining ^ths was held in shares; what must be the average weekly receipts so as to pay the shareholders 6 per cent., the expenses of working the railroad being 45 per cent, of the gross receipts ? X. 1. Explain the terms Par of Exchange, Course of Exchange, Simple and Compound Arbitration. MISCELLANEOUS QUESTIONS AND EXAMPLES. 295 The exchange between London and Paris is 25*5 francs per pound sterling; between Paris and Amsterdam is 117 francs for 55 florins; be- tween Amsterdam and Hamburgh is 11 florins for 13 marks; wliat is the excliange between London and Hamburgli ? 2. (a) Find a sum of money which shall be the same fraction of £69. 9*. 6d. that 2 cwt. 2 qrs. 10 lbs. is of 36 cwt. I qr. (/3) Reduce 12-^\ O'jd. to the decimal of half-a-guinea ; of £1; of £1000; of £-000001. (7) Divide 1255 by 1'004; 1255 by 1004; -012550 by 1004000; and multiply '123 by 3-4343. 3. (a) What sum must A bequeath to B so that B may receive £1000, after a legacy duty of 10 per cent, has been deducted ? (/3) In what time will £2500 double itself at 4 per cent, simple interest ? (7) Shew that the interest obtained by investing a sum of money in the 3 per Cents, at 82i- is to the interest obtained by investing the same sum in the 3^- per Cents, at 93^, as 34 is to 85. 4. If the price of 100 bricks, of which the length, breadth, and thick- ness are 16, 8, and 10 in. respectively, be 5.9. 4(1, v>hat will be the price of 9760 bricks which are one-fourth less in every dimension ? 5. A contractor sends in a tender of £5000 for a certain work ; a second sends in a tender of £4850, but stipulates to be paid £500 every three months; find the difference of the tenders, supposing the work in both cases to be finished in two years, aud money to be worth 4 per cent, simple interest. 6. A railway train travels 27 miles per hour, including stoppages, and 30 miles per hour when it does not stop; in what distance will it lose 20' by stopping? 7. If 2 boys and 1 man do a piece of work in 4 hours, and 2 men and 1 ])oy can do the same in 3 hours ; find in what time a man, a boy, and a nian and a boy together, respectively, can do the same. 8. A field is 300 yds. long and 200 yds. broad; find the distance from corner to corner. If a belt of trees 30 yds. wide be planted round it, find the area of the interior space. 9. A boy can buy at a fruiterer's either 2 cocoa-nuts, or 12 dozen filberts. He buys the cocoa-nuts, ana then commences a series of ex- changes, obtaining 5 pears for a cocoa-nut, 5 apples for 2 pears, 2 oranges for 3 apples, 21 hazel-nuts for an orange. 2 filberts for 5 hazel-nuts; is he better or worse off than if he had bought the filberts at the fruiterer's? 296 ARITHMETIC. 10. A grocer buys 48 lbs. of coffee at 10c?. a lb., and mixes it with 12 lbs. of chicory which cost him 3*. id. ; what will be his gain per cent, if he sell it at ISd. per lb. ? 11. Capital originally invested so as to yield an annual income of £4500 at the rate of 4| per cent, is re-invested at 5 per cent., and then divided among 3 persons in shares which are as 4, 7 and 9. What is the yearly income of each ? 12. Riding a journey of 27 miles into town, I meet the coach, which left town at the same moment that I started from home (7 o'clock), at the 18th milestone from town. Supposing that it travels 10 miles an hour, determine the hour when we meet, and the time when (proceeding at the same rate as before) I shall reach London. XI. 1. What is meant by discounting a bill? What is meant by the " three days of grace" ? What does a Banker gain by discounting on July 1st a bill of £150 dated May 22nd at 3 months at 4^- per cent ? 2. (a) A lb. of powder costs 3?., and the charge for a gun is 2} drams, how many shots will 6s. 9d. worth of powder furnish ? (/3) Wheat being 42*. a qr., calculate its price per hectolitre in French money, supposing a hectolitre = 22 gallons, and the exchange to be £1 = 25 fr. 30 c. 3. A cube contains 2*876 cubic yds. How many linear feet are there in (1) an edge, (2) a diagonal ? and what is the area of one of its faces ? 4. The cost of publishing 1000 copies of an English work in two volumes is 500 guineas. What is the cost of publishing 1500 copies of a French translation of it in three volumes, each volume of the translation costing as much as a volume of the original ? 5. Of two men, one works regularly 7 hours each day in the week, the other does no work 2 days in the week, but endeavours to make up by working 3 hours per day for 2 days, and 12 hours per day for the other two ; how many days according to his rate of work does the former gain in a year ? 6. A, B and C having equal shares in a ship, sell respectively one- half, one-third, and one-quarter of their shares to D, who dies and leaves his share equally among them. If 5's and C's interest in the ship be then worth £7782. 1*. 8d. what is the value of ^'s share ? MISCELLANEOUS QUESTIONS AND EXAMPLES. 297 7. The difference between the interest of a certain sum for one year, and the discount on the same sum due a year hence at 5 per cent, is £1 ; find the sum. 8. How many deal planks each 10 ft. long, 11 in. wide, and 2irin. thick are required to plank a floor 20 ft. 2 in. wide, and 30 ft. long ; and what is the cost of the timber at £1 per load of 50 cubic feet, and £\ per load for sawing and carting ? 9. The solid contents of a sphere being | of f f f of a cube, the side of which is the radius of the sphere, and a cubic foot of iron weighing 450 lbs; find the diameter (in inches and tenths of an inch) of a 681b. cannon-ball. 10. The distance from Aio B is 12 miles, 2 miles of which is uphill, and 3 downhill ; find the difference between the times in which a person would ride from A to B and back again respectively supposing his pace uphill to be 4 miles, downhill 5 miles, and on level ground 10 miles per hour. 11. At what time between 11 and 12 o'clock are the hour and minute-hands of a watch 1st together, 2nd at right angles, 8rd directly opposite ? 12. I have shares amounting to f th of a property worth £126. 14s. Id., and after purchasing additional shares worth ^th of my own, I sell j^th of my whole interest in the property. What share have I left, and what is it worth ? Express both results in decimals. XIT. 1- What is the general object of a question in the Rule of Three? How does the Direct Rule of Three differ from the Inverse ? How does Simple Proportion differ from Compound Proportion 1 (a) If a garrison of 600 men have provisions for 5 weeks, allowing each man 12 oz. per day, how many men can be maintained for 10 weeks by the same quantity, if each man is limited to 8 oz. a day ? (/3) If a certain number of workmen can do a piece of work in 25 days, in what time will If of that number of men do a piece of work twice as great, supposing 2 of the first set can do as much work in an hour as 3 of the second set can do in 1| hours, and that the second set work half as long a day as the first set ? 2. The amount of a certain sum with simple interest for 20 years is £395. 95. and with simple interest for 10 years more is £461. 7*. 2f/, find the sum, and the rate per cent, per annum at which interest is reckoned. 298 ARITHMETIC. 8. Find the squares of 1039681 and 828776 ; and divide the greater result by the less, to the first significant figure in the decimal places. 4. If one watch loses and another gains at the rate of 1 min. a day, and they are both set at noon on Monday, what time will be indicated by the latter, when the former pomts to 10 h. 49 Jj- min. p.m. on the following Saturday ? 5. The area of one end of a cubical cistern is 12^ ft.; express its capacity in feet and inches. Supposing it provided with two spouts which would fill it in 10 and 12 minutes respectively, and with a tap which would empty it in 15 minutes; what portion of it will be filled by leaving all three open for 5 minutes ? 6. A merchant sells a certain quantity of com at 4G.9. a quarter ; the purchaser on selling again at a rise of 2s. a quarter realizes £15 by the transaction ; how many quarters were sold ? 7. If A possess ^th part of a ship, whose value is £6800, and -B | of the remainder, what should the third partner C pay them for their joint shares to make a profit of 10 per cent, by his purchase ? 8. A person buys 1000 qrs. of wheat at 54-?. per quarter ; he keeps it 7 months, during which time it loses in quantity 2<^ per cent. ; if money be worth 5 per cent, and his incidental expenses be £20, what does he gain or lose by selling the wheat at hUs. a quarter ? 9. A can mow 2^ acres in 4^ days, and B 2^ acres in 3| days ; they mow together a field of 10 acres. How long will it take them to do it, and how many acres will each mow ? 10. 1 kilogramme —10 hectogi-ammes =100 decagrammes =1000 grammes. Find the value of 57 kilogr. 8 decagr. 4 gram, of any article which cost £17 5 fl. 7 c. per kilogramme. Express the result in the English coinage ? 11. The first of six boys can copy 3 lines as soon as the second can copy 2 ; the second 5 as soon as the third 6 ; the third 7 as soon as the fourth 8; the fourth 9 as soon as the fifth 10 ; and the fifth 15 as soon as the sixth 14 ; how many lines will the sixth copy whilst the first is copying 135 lines? 12. A company is formed in which the liability of each partner is limited to the amount of his shares. There are 500 shares of £10 each ; after 3 calls have been made of £2 on a share, it is found that the concern is a failure, and its affairs are wound up. At this period its assets amount to £10217. 0*. O'ld. and its liabilities to £15763. 17*. 6d How much will the company be able to pay in the pound after all the remaining calls are paid up ? APPENDIX. MISCELLANEOUS PAPERS. I. ^ 1. Shew hoW to divide 4 things of the same size and material among 3 chil- dren. A, B, and C, by merely breaking one of the four, and so that ^'s share shall 1 ' \ 3 7 3 be - H 1- - of a whole one, 5'3 share - + 77; + ::;7 of a whole one, and C's share 5 6 2 lU 20 the remainder. 2. A person employs 25 men, and 20 women, who work respectively 12 and 10 hours a-day during five days of the week, and half tim6 on the remaining day; each man receives 3d., and each woman 2rf. an hour. What is the whole expense of labour during a yearl (a year = 52 weeks). 3. If 144 men can dig- a trench 40yds. long-, 1 ft. 6 in. broad, and 48 ft. deep, in 3 dnys of 10 hours each; how long must another trench 5ft. deep and 2ft. 3in. broad be, in order that 51 men may dig it in 15 days of 9 hours each ? \^4. Explain what is meant by compound interest. What is the difference be- IweeKtbe simple and compound interest of £345. 5/f., for 2 years, at 3*5 per cent.? 5. The length of a rectang-ular field which contains 2 acres, 3 roods, 5 poles, is 151 yards, 9 in. ; find its breadth. 3/„, ^. . , , ,5030-912 Which is the g-reater V2 or \'3? Find the cube root of 65536 7. What is the worth of 16 lbs. of a mixture of tea which Contains b\ parts of black worth 4j. 8(i. per lb,, and 4^ parts of green worth Qs. per lb,, and 2^ parts of orange pekoe worth 3s. Qd, per lb. 7 8. Find the equated time of payment of £200 due 14 months hence, and of £300 due 19 months hence ; and determine the present value of the whole sum (supposed to be due at the equated time) allowing 3| per cent, simple interest. 9. Supposing the supply from California to become so great that the market price of gold decreases in the ratio of 7 : 5, what would be the absolute loss sus- tained by a fundholder upon every £100 which was paid off at par, if he had bought in before the depreciation took place, when the price of stock was at 89^ 1 10. A person buys tea at 6*. a lb. and also some at 4^. a lb. In what propor- tions must he mix them, so that selling his tea at 5s. ^d. a lb-, he may gain 20 per cent, on each lb. sold? 11. Find the cost of papering a room 19 ft. Bin. wide, ^4^. 4 in. long, and 13^ ft. high, with paper 2f ft. wide, which costs lis, per piece of 12 yards; the windows, 5pd parts not requirmg paper, making up a sixth of the whole surface, 300 APPENDIX. 12. A certain sum produces an annual income of £200 when invested in the 3 per cent, consols, and of £260 when invested in railway 4 per cent, preference shares at par. Required the sum invested, and the price of consols. II. 1. Explain the method of pointing in extracting the square roots of whole numbers and decimals. Find \/ (57 2 14096), and also, as far as three places decimals, V(572- 14096). 2. What kinds of questions can be solved by means of the Rule of Three! Distinguish between the Rule of Three Direct, Inverse, and Double. 3. A bankrupt pays 5/^. 7c. 5/n. in the £, what ought a creditor to receive On adebtof £1920. 7//. 5c.? 4. A person, after paying from his rental Id. in the £. for income-tax, and 3^ per cent, on a mortgage of £4000, has £1568. 13s. Ad. remaining : what was his rental ? 5. On the price of 25 vols., bought at 3s. a vol., the bidder is allowed 5 per cent.; on that of 12 others, at 5^. 3rf. a vol., 11 per cent.; 2^ per cent, of the auction-duty is also paid by the purchaser: what will the books cost ] 6. A person buys 3^ cwt. of tea at 5*. A^d. per lb. and 4| cwt. of tea at 3s. 2H. per lb., and mixes them ; he sells 5 cwt. at 4s. Qd. per lb. : at what rate per lb. must he sell the remainder so as to gain 20 per cent, on his outlay 1 7. If 2 cub. in. of iron weigh as much as 15 cub. in. of water, and a cub. ft. of water weigh 1000 oz.; find the weight of a cubic yard of iron. 8. Three horses do the same work as 5 ponies, and 12 horses can just draw a certain load ; how many ponies would be necessary to draw half the load 1 9. A after doing ^ths of a piece of work in 30 days, calls to his assistance B, tind together they finish it in 6 days ; in what time would each do it separately ? 10. What is the difference between Interest and Discount ? A person purchased land at £60 and £56 per acre respectively ; the former he let at £2, and the latter at £2. 2s. per acre per annum : find the rate of mterest he obtained in each case, and the advantage of the second purchase over the first. 11. In a certain lake the tip of a bud of lotus was seen a span above the sur- face of the water. Forced by the wind it gradually advanced, and was submeiged at a distance of two cubits. Compute the depth of the water. 12. A, B, and C are partners ; A receives ^ profits, and B twice as much as C, find the capital of C, ^'s income being diminished £40 by a fall of ^ per cent, in the rate of profit. 13. A man expends £1000 in the purchase of Great Nugget shares of £5 when they are at 2 premmm, and £500 in the purchase of Agua Frias of £2, when they are at f discount, he sells out again when the Nuggets fall to par, and the Agua Frias rise to 3 premium. What does he gam or lose after paying the broker I per cent, on all the money which passes through his hands? APPENDIX. 301 III, 1. Define a Vulgar Fraction. How many kinds of Vulgar Fractions are there"? Shew that multiplying- the numerator of a fraction by any number is the same in effect as dividing the denominator by it. Simplify 2. If 1 lb. Avoirdupois be equivalent to 7000 grains Troy, and 66 shillings weigh 1 lb. 'iroy, find the value of 20 avoirdupois ounces of silver. 3. A gentleman dying- leaves property worth £23,100 among- 3 sons and 4 daughters, directing that the sons shall have alike ^ more than their eldest sister, who should have £300 more than either of her younger sisters, they sharing alike. How much did each get "? 4. Find at what rate simple interest a sum of money would amount in 2 years to the same as at 4 per cent, compound interest. 5. (1) The edge of a cubical beam is 18 inches; what is the edge of one contain- ing 8 times as much ? (2) Find the side of a square field containing 2ac. 121 yds. 6. If 5 men and 7 boys can reap a field of corn of 125 acres in 15 days, in how many days will 10 men and 3 boys reap a field of corn of 75 acres, each boy's work being ^ of a man's] 7. A person invests £962. 10s. in the 3 per cents, at 77, and when the funds have fallen 1 per cent, he transfers his capital to the 4 per cents, at 95; find the alteration in his income. 8. Which is the more profitable investment; the purchase of 3 per cent, con- sols at £96, or the purchase of shares in an insurance oflfice at £227 per share, the annual dividend on a share being £7. 10s. ? 9. If the wholesale dealer sell to a retailer at 10 per cent, profit, and the retailer sell to the consumer at 50 per cent, profit, what proportion of the price paid by the consumer is profit? 10. Find the area of a court-yard 9 yds. 2 ft. 6 in. in length, and 7 yds. 1 ft. 8 in. in breadth, by duodecimals, and explain your method of operation. 11. A wine-merchant pays £70 for a pipe of wine, and bottles it oflT into an equal number of quart, pint, and half-pint bottles. How many dozen of each has he, and at what must he sell it per dozen to gain 15 per cent, on his outlay! 12. With a gallon of rum which cost 15s. a man mixes a quart of water, and then sells it for 16s. a gallon: with a gallon of gin at Us. he mixes 2.^ pints of water and sells it at 12s. a gallon : and with a gallon of brandy which cost 22s. he mixes 3 pints of waier, and then sells it for 23s. a gallon ; how much does he gain per cent, supposing him to sell twice as much rum as gin, and twice as much gin as brandy I 802 APPENDlXo IV. 1. Find the square root of '000961, and prove the correctness of the result obtained. What is the length in inches of the side of a cubical box which contains •000027 cubic yards? 2. State the rules for multiplication and division of decimals. Divide 2'50892806 by 92*41035 to four places of decimals, and shew by fractions that the result is correct. 3. Define interest, simple and compound, present worth, and discount. (1) What sum of money lent at 5 per cent, simple interest for 3 years will amount to £828 ? (2) Find the present worth on £487. 5//. 2 c 5 m. due 175 days hence at 3-75 per cent, per annum. 4. A tenant holds a farm of 350 acres, subject to a tax of 3s. 6d. per acre and a corn-rent of 100 qrs. of wheat, barley, oats, and beans respectively. Find the amount of his rent when the average prices of wheat, barley, oats, and beans per quarter are 385. 9d., 21s. Ad., Ms. Ad., and 33s. \Qd. respectively. 5. It is observed that 20 men, all of equal streng-th, build a wall 15 ft. high, 30ft. long, in 60 days, and 35 others, also of equal strength, build a wall 20 ft. high, and 40 ft. long, in 64 days ; what is the ratio of the streng'th of the men of the two classes'? 6. Wliat ought to be the value of £135 in the 4^ per cents., when the 3 per cents, are at 97^ } 7. Standard gold being coined at the rate of £3. 17*. \9' 10' 5- (2^ 8' 6' 4' 2- (') r^r 1-2' S^^B" ^^60' 21' 12' 16* ^^ 26' 11' 13' 7' 22' C6)2of|of4,iof|cf5,li, loflof43. (9) li 5. 1 ' 56' 28' ^^^'^ 9' 22' 18' 11 ' 5 36' (11) lA. '^, 401 113 51 448' 152' 76* <.-2)f,3i,5 ;of9f,^of^of|. 2. (I)?andi. (2)Sandf^. Ex. XIX. (p. 47.) The suras will be : 1. (1) lij. (2)|. (3)1|^ (4)1. <- (6) IH- 29 (7) 36- (8)1. (9)2M. (10) ||. . (11) ll5^. (12) 14f. 2. (1)23^. (2) l3^. (3) 1. '^)i- (5) l#j. (6) 2^2. (7) m- (8) m- (9) IiVj. (10) 15|f. (11) 10^. (12) 5}|^. (13) 3i*^. (14) li^cyV (15) 2^ (16) \m- (17) 72»r. (18) 585f. (19) 444|. (20) 2548^^, 3. (1) liWfi^. (2) 4fi^. (3) 2fMi. (4) 11A-, (5) 231H. (6) 13i^. (7) IBgi^. (8) 8^. (9) 5976. (10) 3tY«. ANSWERS (pp. 49-58.) 321 Ex. XX. (p. 49.) 1.0)1. (2)§- (^)9l- (^)4- <^)^- <«)n- (7) It^. (8) 4jh' (9) 1^. (10) 3if^. (11) 3,^. (12) Uj\. (13) 13^1. (14) 64l|Mf. (15) 19^1. (16) 121. (17) 31^}. (18) ^. ' (19) |. (20) 1. 2. By I- 3. 3t|, 2f f. 4. 10H|. 5. 2^^ 6. The sum of the fractions is 5 times as great as their difference. Ex. XXL (p. 62.) ..«!?. <^)1- (3)J. (.)|. (5)1. (6) 21. (7)10^. (8)|. (9)40. (10) ^. (11) 1. (^2^152- ^''^'i- (^^)2- 2. (1) |. (2)4^. (3) 32^. (4)242l|. (5)1. Ex. XXII. (p. 64.) («)l- 1. (1)4. (2) 2^. (3) 1 33. (4) 1^. ^^)^- (6)^. (^>|- (^)S- (9)3l2- (10) i. (11) 3^. (12) 153. (13) 347f. (14) ]^. ^- ^- 320 496* 3. (1)H. (2)^. (3)5i. (4)5f. (5)|. (6)36. P) 7^. Ex. XXIII. (p. 68.) I. 2. 4^and3x^:8i. 3. (1)37|. (2)8|f. (3)g. (4)4f|-. (5) 3JyV 5. S^f . n. 2. 2lf and 3^. 5. 16, 91 322 ANSWERS (pp. 59-64.) m. 2. (1)5000. (2>^. (3)2. (4) Igfj. 3. I of 4 IS greater by ^. 4. -— . 5. l^V IV. 1 3^. 2. (1)|. (2)j|-,. (3)|I. (4) If. (5),2A. 3. 2S« and ±. 4. i. 5. The quotient is 144 times as large as the produce. ( 33§J- ^'''896' 2IT2* ^^'5' 11 ' • ^^25' 175* ^'^''2240' 184* ^^^^80' 55* ^ ^M08 * W ^^^^2304' 81' ^^^^6' 4- ^^^^1863' 25' ^^^^ lOUO' T5625- r> /,N 6 ,^,1 ,^, 108 ,,, 28 ,^, 20 ,^, 7 ^•^^)49- (2) 160- (^)l25- (^)8-I- (')¥• (^)540»- 2J13 .8.2822400 39 U Ex. L. (p. 146.) o II 1. ;tp of a crown is the greatest, £— the next, and — of a guinea is th3 00 1 y Zu least. 2. The first two are equal, the third less. 3. r^ of a day by 7. ^. ' 1600 Ex. LI. (p. 150.) 1. (1) 9*.; 3s.4ld.', 17s. 6|d. -9908?. (2) 5s.7id.; 155. ll-088d.; 14s. 4^^. (3) £5. Os. l^d. ; £3. 17s. 6^.; Us. ' (4) W3id. ; 7s.7id. -48?.; 2s. lid. (5) Is. 2^d. -1657049., • £2. 15s. 3id. -8128?.; U^.S^d. (6) £3. Os. Hid. -049.; £4. 5s.; OS. 9-12(i. (7) 19s. Bid- ; 7s. Old. ; £2. 17s. 3ld. (8) £1. 18s. 9|d. -192?. ; £3. 4s. 2d.; £\.7s.7d. (9) 2m., 1100yds. ; 2 d., 12 hrs., 55'. 21"; 7 oz., 4dwts. (10) £24. 12s. 6-24999936d. ; 4ilbs. (11) 3 qrs., 11 lbs., 4-56192 oz.; 81bs., 3-2G4oz. ; 14 po., 2yds., 7-2 in. (12) 4tons, 3cwt., Iqr., 61bs.,2-56oz. ; 3cwt., 2qrs., 14]bs. ; 8sq. po. (13) 3 lbs., 10 oz., 5*568 grs. ; 2 qrs., 3 bus., 3 pks. ; 14 c wt., 12 lbs., 12-96 oj (14) 3ac., 3ro., 14po.; 63gals. (15) 37p0o ; 9d., IShrs. (16) £506. 6s. 6d. ; 19 qrs. (17) 7 ac, 3 ro., 20 po. ; £3. Is. 7td, (10 £5. 0s.3ad.; 2 m., 1150 yds., 2-052 ft. (19) 13 sq. yds., 1 sq.ft., lll-6sq. in. ; £126. Is. 2d. -2465g. (20) 4m.,6po., 1yd., 2ft., 11-97696 in.; £15. 16s. 6-^d. -6^, 1- ^ ^s' ^• 92 135 ^' 840 «•!• 9. ii. 400 ,« 233 10. 20- ANSWERS (pp. 151-160.) 333 2. (1) 7s. 8rf. ; 9s. 5d. ; 5oz., I2dnts., 16 grs. (2) I5s. 6d. ; Is. 51(1. ; \3s. 4d. (3) 6s.', £2.\6s.2d. (4) £6. 13s. ; £9. 10s. 3^d. (5) 6 sq. yds., 108 sq. in. ; 3 fur., 10 po., 3 yds., 2 ft. ; 20 d., 6 hrs. (6) 8xVl>ac.3 20 hrs., 30 mi. 3. 7s.;10s. 6ii. 4. £1. 10s. 5. (I) £2. \6s. S^d, (2) £83.7s.2f(/. (3) £\.2s.9^d. (4) 5s. 7-012J. (5) 152wks., 5d., 10hrs.,543^sec. 6. -0231 of a guinea. (6) 1 TO., 39 po., 28^ sq. yds., -gfQ sq. m. Ex. LII. (p. 154.) 1. (1) -316; -435416. (2) -23125; -796875. (4) -503125; -0572916. (5) -08472 ■ -57738095*2. (7) 1-35; 1-46875. (8) 3-590625; 1-43625. (3) -675; -003125. (6) -375; -694. (9) -2232142857; -857142. (11) 350-9P; 1-32531 (13) -034375; -30016741 (15) -2785493827166; -875. (17) -67857142 ; -00002546296. (19) -000015...; 5-857142. (21) 1-916; 14-24. (23) 75-789 ; 5212-307692. 2. (1) -45; 2-i42857. ._ (4) -225; -511. (10) 1-365; -0000625. (12) -22083; 48-083. (14) -273-29545; -07-2916. (16) -4027; -61875. (18) -93; -8-2285714. (20) -0334821 ; 82-5. (22) 114-54; -00061 (24) -01875; -805; -7317. (2) -148809523; -3. (3) -28125; -013671875. (5) -00243...; -000080... (6) -000304...; -065625. (7) -288; -546875. 3. -3821 4. -0475. 5. -11825396. 6. -1027. 7. 1-694 8. 2s. dd., -45. 9. -3140625. 10. (1) 2c. 5m. (2) 4 c. l|m. (3) Ic. 8|m. (4) 2'fl. 5c. (5) 5fl. 2 c. 5 m. (6) 8fl. (7) £5. 6fl. 2c. 5ra. (8) £54.3|fl. (9) £-20. 9fl. 8 c. l^m. (10) 7 fl. 6 c. 9-7916 m. (11) 7fl.3c. 4m. (12) £'2. 7fl. 9c. 6^m. (13) £3. 4c. 9m. Ex. LIII. (p. 160.) 1. £80. 12s. 6d. ; £36. 4s. 6d, 2. £62. 8s. 6d. ; £5. 14s. 3. £9. 15s. ; £35. 7s. O^d. 4. £271. 5s. Ad.; £172. 15s. 104 5. £17. 8s. 4d. ; £10. Is. 11^^. 6. £234. 12s. ; £927. 13s. 6d. 7. £106. 17s. 64. ; £2600. 19s. 10a. 8. £651. 4j. 6rf. ; lis. 334 ANSWERS (^pp. 160-171.) £254. 7s. 6d. ; £6899. Us. Id. 10. £170. 13j. 2f(f.; £790. I7s. 9d. £2005. 16s. 8^.; £669. 3s. 9d. 12. £837. 3s. ll^d.; £398. 13. 5^d. £14842. i6s.; £1737. 18s. d^d. 14. £720. 3s. 4^d. ; £2358. 16s. Q^d. £27820. 18s. 3d. ; £82654. 9s. \d. 16. £39061 . 8s. S^d. ; £247968. 12s. b^d. £250014. 13s. 8|rf. ; £44359. Is. 10|d, £16982. 14s. 7^d. ; £4981. Is. l|d. 19. £1075. 8s. 8d. ; £2173. Is. 9d. £552. 14*. 5Jd. ; £981. 19s. Id. 21. £1927. 16s. lO^d. ; £1903. 18s. 8^d. f ^. £11342. 13s. 5^d.; £5498. 9s. lO^d. yt?- £69241, 9*. l-^d.; £24617. Os. \^d. }%q. £19. 7s. 5|d. ^q.; £144. 14s. 0|J. ^g. £51. 12s. 9^d. ^q. ; £986. 17s. lOd. £1775. 9s. S^d. §7. ; £831. 3s. 2ld. \q. £10689. 17s. 8id. \q.; £12126. 7s. ll^rf. §^. £12. 15s. ll|rf. 29. £215. 16s. 8|d. 30. £89. 6s. l^i. £467. Is. 6|d. ^q. 32. £12. 5s. lO^d. ^q. 33. £2. 15s. \\\d. £147. 16s. ll^d. ^g. 35. £230. 16s. 8^ci. ^g. 36. £4. 9s. S^d. ^<7. £62. 8s. Qf^d. 38. £14. 19s. lO^d. 39. £595. Qs. ll|d. ^g, £5].8s.0^rf. 41. £308. 6s. 8|d. |§^. 42. £33. 8s. 3|d. Jf^g. £8. 3s. Aid. Iq. ' ^ Ex. LIY. (p. 170.) I. 3 sq. yds.,! sq.ft., 60 sq.m. 2. 238 sq.ft., 90 sq.m. 3. 31 sq.ft., 87 sq.m, 4. 52|fft. 5. 32ft. 6. 17ft., Sin. 7. 90iVplank3. 8. 683sq.yds., 2sq.ft., 25sq.iil. 9. £62. 5s. 5d. 10. £3. Is. 5i. II. £3. 15s.6^d. 12. (1) £11. 15s. (2) £11. 6s. 5rf. (3) £5.0s. 7^d. 13. (1) 273 sq. ft., 63 sq. in. (2) 396 sq. ft., 60 sq. in. (3) 144 sq. fc, 8'. 5". 3"'. (4) 377 sq. ft., 10'. 5". 7'". 6"". (5) 42sq. ft., 6'. 6". 4'''. U"". (6) 2893 sq.ft., 8'. 4". 10". (7) 274 sq.ft., 4'. 10". 9"". 14. 221^ yds. 15. 26yds., Oft., 4 in. 16. £10. 17. (1) £5.2s. 9^(f. (2) £l3.ls.6id.lq. (3) £12. 12i. 6|%7. 47. £175.7s. 7i(/. 48. £14. 14s. 49. £100. 9s. 0|J. 51. £1682. 52. £2110. 6s. 3d, 53. lOh. 40'. 36q:V'. 70 ft. 8-232 in. 55. 9s.Shd.; £58. 13s. 3cZ. 56. 4^(5% cvvt. £747. 10s. 58. £4. 10s. 59. 5s. 6d. 60. 15 hrs. £40. 8s. 3d, 62. £157.10^. 63. 208 days. 64. £1. 9fl.5c. 10| days. 65. 65/Vyd3. 67. He loses 3i(Z, 5foVV before 4 o'clock. 69. £760. 70. 70 days. 41661 yds. 72. 240000 lbs. 73. 2240|fff. 74. £94. 10s. 168 lbs. 76. Monday fortnight, at 6h. 36m., p.m. 77. 72 yds. 13.". 4d. 79. 466650 lbs. 80. 100 days. 81. £26000. £18. 10s. 3d., nearly. 83. 80 days. 84. £211. 19*. 3^/. £4. 4^. 86. 9^ yds. 87. £39. 9s. 2id. f g. 88. 207 : 82. 3 days. Ex. LVII. (p. 224.) 15 men. 2. 7 men. 3. 66 days. 4. 7200 soldiers. 19/T7bus. 6. 600 ac. 7. 18iff-mi. 8. 9Hcwt. £326. 13s. 4d. 10. 11 mo. 11. £6l.lSs.5d. 11 cwt., 3 qrs ., 141b3. 13. 12hrs. 14. 8.^ wks. 15. £20. 3d., 6hrs. 17. lOhrs. 18. £50. 8s. 9c?. 19. 500 reams. 9 days. 21. 8wks. 22. 64 days. 23. 350 men. 2400 men. 25. 47 tons, 17 cwt., 66 lbs. 26. 324 men. 48 days. 28. £332. 5s. 2j_V. 29. 2268 cub. ft 30. 13s. 4^;. 8 ft. 32. £2. 3s. IJrf. 33. 1320 yds. 34. 18|fft. £97. Os. 4id. a§. 35 days. 37. 19-36 days. 38. 49-3 lbs. £509. 12s. 22 8S8 ANSWERS (pp. 230-239.) Ex. LVIII. (p. 230.) (2) £12.85. (3)' £45. (4) £71. 5s. (6) £167. (7) £75. 12*. 9}J. fg. (8) £6. 18s. lOd. (10) £1. 4s. 8ld. iq. (2) £3215. 16s. M. (3) £1318. 5fl. 3c. 4|m. (5) £2249. 13s. 21 J. If g. (6) £1615. 6s. lO^d. ^^7. (8) £1939. 7c. 3-8281 25 m. 3. (1) £48. 2s. 6d, ; £423. 2s. 6d. (2) £72. lis. lid. ; £519. Is. l^d. (3) £4. 16s. 3rf.; £224. 165.3d. (4) £27. 19s. 0-}^d. j £271. 9s. 0^^. (5) 2s.4yW5rf-; £10. 12s. 4^^^. (6) £54. 3s. OtVo^. ; £739. Is. Sx^^i. (7) £1. 2s. 67V. ; £43. 2s. 67V- (8) £31. 18s. l^d. ||f 5. ; £352.13s.7id. flf^. (9) 6s. IIJ. nearly; £34. 16s. lir/. nearly. 4. £32. 4s. Old, 5. £20. 6. £2000. 7. £567. lis. 4^rf. 1. (1) £4. 5s. (5) £74. 18s. 6d. (9) £17. 14s. 6d 2. (1) £1085. (4) £\66.l0s.0}2d.^rj. (7) £416. 10s. Sy. i^q 1. £125. 6s. Bd. 4. 4| per cent. 7. £560. 10. 5§ years. 13. £315. 10s. 8d.; 2^ years Ex. LIX. (p. 232.) 2. 4 per cent. 5. 17 yrs.' 8. £91. 13s. 4d. 11. 4| per cent. 3. 5 years. 6. 3^ per cent;. 9. £345. 17*. 6d. 12. 20 years. 14. £225. 1. 4. 7. 10. £163. 4s. £787. 8s. 1 JJ^^^cf. £16. 8s. Oid. nearly. 4s. 2^(i. ^q. Ex. LX. (p. 235.) 13. £90. 14^x5. £893. 8s. 4^d. 3. £267.2s. 5icf. Y§^7. 6. £1942. 4s. 9W nearly. 9. £350. 8s. 5d. 12. £240. 15. £16. 8s. lOiJ.^.7. £1. 5s. lid. nearly. £714. 8s. 7 id. If?. £205. 8s. 2^5^^^. £1-578528. Ex. LXI. (p. 239.) 1. (1) £270. (2) £245. (3) £666. 13s. 4d. (4) £280. (5) £450. (6) £382. 14s. lOjfntd. (7) £558. Os. llffd. (8) £1260. (9) £34. 9s. liUd- (10) £1239. 3s. \^}d. (11) £2000. (12) £262. As. 5ld. ^q. (13) £765. (14) £ 162. 9s. 5^W- (15) £400. (16) £1953. 2s. 6d. ANSWERS (pp. 239-249.) 339 2. (1) 165. Sd. (4) £\.\s.9UTd. (8) £48. 9s. (11) 2s. Ud. nearly. (15) £2.6s.8d. (2) £30. 7^. 6d. (3) £2. 12s. 3fld. (5) £7. (6) ^5. lis. 5* ^/. (7) £140. (9) £16. lis. 8|ff /. (10) £1. Os. 6^^\%d. (12) 15s. ditjWfd' per c.e°t. (14) 20 per cent. Ex. LXII. (p. 243.) 1. (1) £3800. (6) £597. Os. 3^d. (10) £2600. 2. (1) £2418. (5) £3040. 4s. 9|d. (8) £972. 10s. £4064. 14s. 7hd. 3. (1) £36. (2) £240. (5) £67. 15^. ll^d. (8) £111.8s. 1-^ffc/. <9) 4. (1) £1700. (2) (5) £2164. 2s. 6d. (6) 5. (1) £3. 5s. lllfi. (4) £4. 17s. li^srf. 6. £244. 13s. (2) £800. (3) £525. (4) £950. (5) £4300. (7) £1059. 12jfjs. (8) £5050. (9) £2234^2^. (11) £3091. 10s. 2M^. (12) £10566. 0s.9#3rf. (2) £1488. (3) £2775. (4) £1831, (6) £1444. 105. 2.Vi.f^. (7) £2450. 6s. (3) £55. 6f]. 8c. 7^m. (6) £112. O.s. 3l|f(/. £178. 15s. 2^V- £6432. (3) £1800. £875. 7s. 93d. (2) £3. 14s. 5|fi. (3) (4) £78. 15s. (7) £159. 12s. (4) £1739. £4. 13s. l5f|:f. II. 14. 17. 19. 21. 23. 26. 28. 31. £66. 9s. 9^d. 12. £21. 5s. £5. 13s. Ifgif/. £378; £11970. The 4 per cents. 1 he 3^ per cents £1666. 13^. 4d. The 3 per cents. £7900; £6310. 2s. 6c/ £25; £22. 11*. 71}^- 7. £3. 6s. 8d. ; £3. 15s. ; 8s. 4d. 9. 77^; £1542. 17s. l^-d. 10. £1000. £1575. 13. £104. 8s. 4d. 15. £2. 10s. 16. £175; £170. 9s. IjV- 18. £2729t^; £5. 10s. 3i^rf. 20. £6. 5s. \0-^jd. 22. £7. 15s. 4d. 24. The Railway Shares. 25. £22. 2s. 6d. £13397. 10s.; £13980. 27. £12319. 6s. 3^%-5rf. 29. £32. 5s. 30. £86. 32. £20,000; £225000. Ex. LXIII. (p. 249.) 1. £2. 10s. IH. 2. 8s. 8c/. 3. £78. Os. lOJ. 4. He gains £15. 12s. 6J. percent. 5. Ud. 6. £9. 2s. 8|^d 7. £48. 7s. 0|(/.|f^. 8. £10. 14s. 3|d. ■f(/. 9. £10. 5s. 10. £33. 6s. 8(/. 11. £30. 12. 5s.S^d,; 5s.6d, 13. 8s. 9d. 340 A>:swERS (pp. 250-260.) 14. .£37. 1^. ind.^q. 15. £25; £2. 2s. Ud.j\q. 16. £27. 17. 14s. 4|y/. 18. £3. 18s. 19. 4s. 7^./. 20. £27, 21. £648. 7s. 6<7. 22. £3.75.6^.; £14. 5s. 8if/. fg. 23. £96. 7s. 3f id. 24. £16. 25. Is. lO'^d.; 2s. l^d. ^q.\ 3s.9d. 26. £550. 5s. ll^T^rf.; £18. 6s. lOr^eV- 27. £96. r2s. ; 12s. 3^ 28. 30 quarters. 29. 12^ per cent. 30. £40. Ex. LXIV. (p. 255.) 1. 6, 28, and 38. 2. £4. 3s. 9d.; £13. 8^. 3. 516,860,1204, 1B92; £149. Us. 5^p. ; £179. 9s. S^fd.; £170. 18s. o^d. 4. £179. 8s. 8r/.; £142. 9s.; £99. 3s. 4d. 5. 12 cwt., 30^1 lbs.; 3 cwt., 30i\lbs.; 2cvvt., 50l$lbs. 6. £396; £324. 7. 2cwt,, 1 qr., 14Ibs., 6^y oz. 8. £3250; £2166. 13s. 4d.; £1083. 6s. 8d. 9. £350; £450. 10. £12. 10s., £12. 10s., £25, £50. 1 1. ^'s share =£5000, jB's share = £3750, C's share = £3125. 12. £1350. 13. 5^i- months. 14. 4f months. 15. 12 months. 16. £3. 10s. 17." £126. lis. 3d. 18. A ought to have £80, B £90, and C £84. Ex. LXV. (p. 259.) 1. 187-93; 352-4625; 2255 76; 4-285-944; 5639-4; 84872-97. 2. 15-6-25; 23-456...; 8-984375; 2-530...; -048.... 3. -005; -0275; '043; -055-25; -26*3; 2-3005; 5-000138. 4. 1463-65 gal. 5. 8437-5 bus. 6. iff- 7. 10-72...; 8-71.,. ; 10-44.... 8. 5-24...; 20-29...; 16-11.... 9. 1045678-375 persons. 10. 3 of the age of 18 years ; 19 between 15 yrs. and 18 yrs. ; 38 between 12 yrs. and 15 yrs. ; 133 between 10 yrs. and 15 yrs.; and 190 under that age. 11. Weight ofoxygen = 1116-7744 lbs. ; weight of carbon = 969- 1 36 lbs. ; weight of hydrogen = 1540896 lbs. 12. 1712 voters on the register. M obtained 44|?f per cent. ; 5 42ff| per cent. ; A 'i2^j% per cent. ; and H il^^ P^i' cent. 13. He loses £3. 8s. 14. No. of male criminals : no. of female criminals :: 5 : 4. ANSWERS (pp. 263-282.) 341 Ex. LXVL (p. 2G3) 1. 20066.12^.1^^. 2. 2851507 jJ^Y qrs. 3. 9s.9}od. 4. Average ag-e of boys =9f yrs. Average age of g-irls^lO^^yrs. Average age of whole class=10|^f- yrs. 5. 4-447.. .days. 6. 3-9428571. 7, £431. 4s. 9|(/. |g. 8. £372. 185. ijf/. ff^. Ex. LXVII. (p. 2G5.) 1. 35641 francs, 6| centimes. 2. £1271. 13s. 9i^ijrf. 3. 1246 pias, 6ff rials. 4. £566. 135. 4c/. 5. 33s. 4d. 6. lmilree=54f7V. 7. The direct way. 8. £19. 10s. 7.id. ; 25 francs. 9. He gains £11. 5s. 10. Through France. Ex. LXVIII. (p. 273.) 1. (1) 17; 24; 38; 64. (2) 81; 145; 416. (3) 314; 193; 108. (4) 999; 989; 908. (5) 5432; 3789; 2312. (6) 15367; 531441; 16807. (7) 543200; 2039750. 2. (1) 12-96; 5-37; 240-1. (2) -59049; 6-2573. (3) -207; -0374; -0451. (4) 2403; 2-403. (5) 347-6905; 490-304. 3. (1) 4; 1-2649...; -4; -1264 (-2) 15-3492...; -3162...; -1; 2-2360... •7071 (3) -02; -0284...; 19-4901 (4) 4^ ; 12-4007... ; -5773... ; ^. (5) -7745...; -2425...; 1-4719...; -8819 (6) 15-4919...; 1-2^ 4-8062 ; 6-4807 4. -04375; ^^ . 5. 1400 yds. 6. 74^ yds. 7. 256 yds. 8. 28^ yds. 9. 45 yds. 10. 843 ft. 11. -07122 ; 98 yds., 1 ft., 1 in. 12, 90 miles. 13. 475. ..ft. 14. 105. 15. 543-2 yds. Ex. LXIX. Cp. 282.) 1. (1) 12; 15; 31. (2) 38; 48; 67. (3) 88; 93; 9?-. (4) 134; 411; 203. (5) 631; 305; 364. (6) 258; 638 j 975. (7) 3002 6031 2. -73; 3-19} 45-7; -097; -124; -029. 22—3 342 ANSWERS (pp. 282-287.) 3. (1) 1-442...; -669...; '310 (2) ^; ^; 3-546 (3) 7|; 1-930...; 1-442 (4) -046...; -425. 4. 6-16; 1-232. 5. Each edge = 27-2 in. 6. 1369sq.ft. 7. 3ft., lOin. 8. £25. 18s. UJ^d. 9. £lO.Os.ld. 11. 2-8ft. ne-arly. 12. 81; -04 "" Ex. LXX. (p. 283.) I. 2. £2. 8^. 3|d. f g. ; £4. I6s. lid. f g. ; £l. 4s. llirf. h- ; £9. 13s. S^d. f g. 3. (a) FT^; (P) If- 4. £693.2^.10./. 5. 23 boys. 6. 7^ ft. 7. 937 lines; length of remaining line = '02268 in. 8. 488054166 sq. in. 9. £65; He loses £13. 10. £781. 5s.; £913. 19 -048. 11. £3. 17s. 6d. 12. £8000000. II. 1. 1300000507364; Two hundred and thirty-six billions, forty-five thousand nine hundred and seventy-eight millions, two hundred and thirteen thousand, four hundred and seventy-eight. 2. Is.lid.Uq- 3- £4062. 10s.; £12. 7s. 4|J. 2% 5. 4. (a) h; (/3) 3; (7) £6. \6s.5d.; (a) •2256-25; 17-1 in. 5. A ought to receive 2s, 6d ; B, Is. 6d. ; C, 6d. 6. £136. lis. llf^c/. ; £163. 18s. 4|2(/. ; £229. 9s. 8|f d. ; each person ought to pay Is. 1^}^. 7. 72 days. 9. Height = 12 ft. Length = 16 ft. 10. He loses £100. 11. £1306. 10s. 12. £560. in. 1. 1. 2. (a) 1-3010416; (J3) 2-285714. 3. 90 additional men. 4. £10. 14s. 3-^d. 5. 1^ hr. 6. (a) 3-8 ft.; (/3) square = 10260271849; (7) square root = 3253. 7. £448. 8. 3tons. 4cwt. 3qrs. 41bs. 13oz. ' 9. £6666. 13s. 4rf. 10. £1. 11. 32-25. 12. The second is the best investment. IV. 1. 14s. sy. tg.', liis gain is £3. 10s. 2. (a) £1804. 9c. 5m.; (/3) £2764. 8fl. 2*i42857c. 3. 71 days. 4. 189 in. 6. First-class ticket = 10s., Second'class ticket = 6s.8(/.; Rate per mile =2(i. 6. 60 days. 7. £315. ANS^YERS (pp. 287-294.: 343 8. 14d. 9p. Ic. 6fc.s. 9. £5i0.4s.3hdA^lq. 10. £310. 11. 76 days. 12. 624 bales. 1. (a) 3937 yds.; (/?) 166Iirs. 40'. 4. £8. 5s. 9J.; •31571428. 5. £1. 9 fl. 7c. 3*972 m. 6. 1-000049...; -3107.... 7. 75 days. 8. ^'s share = £4224 ; B's share==£3520 ; C's share=£5280; D's share = £3960. 9. £1. 17i. 6ri.,- £7. 10s. 10. 50000000 qrs. 11. 300 stones, 12. (a)8^ll^s.; (/3) From 75 to 60. VI. 2. IjVhr. 3. 2^gros. 4. £1040. 5. 4 per cent. 6. 2 per cent. 7. £2125. 8. (a) ^ : C :: 7 : 13 ; 03) |-i|g i (y) -21; 210; 210000. 9. 448 lbs. 10. 371200 ac. 11. (a) 70; (^) £1388. 175. 9i(i. 12. 190 and 185 votes. 2. 645-12 cub. ft. 3. 6§ pages. 4. 10s. 10§rf. 5. 45 boys. 6. 62h per cent. 7. Kidderminster the cheapest, drug-get the dearest. 8. (a)T^; 1; 08) £1; -1925. 9. (a) 80; (/3) £843. 15s. 10. 12^ percent. 12. A and £ can each do the work in 74j"Ydayp, C can do the work in 157^ days. VIIl. 1. («) H; f ; ^V ; i%V^- (/3) i- (7) ^ : ^ " 64 : 63. 2. £2. lis. 8d. " 3. U miles per hour ; 1 hr. 4. £14. 8s. 9cf. 5. A should have 15s., J5 12s., and the boy 3s. 6. £39. 7. 512|t bus. of wheat, 888f bus. of barley, 1212^ bus. of oats. £630. 13s. 9^^. f|f ^. 8. £\5A9s.0ld. ^ q. 9. -0008997 ; 29 ; 125 gr?. 10. (a) 4-002 , (2) i- (/3) 8-004; (2)3^. (y) 2 ft. 4 ft. 8 ft. 11. 4^ miles. 12. 22iwks. IX. 1. (a) A loses £90, i?, £180, C, £450. O^) ^ ou^ht to have £51. 12s., B £81. 2. 6s. 10^(i. 3. 186 men. 4. £1. 9s. 2^J. i^^?. 5. 12' past 7 o'clock. 6. 39 parcels. 7. (a) £3. 14s. 3a(f. 2V'7. (i3) 4|yrs. (7) 4 per cent. 8. 42592-2588 can read and write; 81808-8102 can do neither; 14516-931 can readonly. 9. 291600 leaves. 10. £40. 11. (a) 405 guns. (/3) £1508. 15s. 7hd. ^q. 12. £4020. 19s. 6|d. If |g. 844 ANSWERS (pp. 294-298.) X. I. 14J marks. 2. (a) £A.\9s.3d. (/3) M47619d; -6025; -0006025; 602500. (y) 1250 ; -0125 ; '0000000125 ; ^^^. 3. (a) £1111.25. 2§i. (^) 25yrs. 4. £10. 19^. Tjd. 5. Difference in tenders = £10. 6. 90 miles. 7. A man will do the piece of work in 7^ hrs,, a boy in 18 hrs., and a man and a boy in 5^ hrs. 8. 360-5... yds. ; area of enclosure = 6ac. 3ro. 30po.22|sq.yds. 9. Worse. 10. 50 per cent. II. £1000; £1750; £2250. 12. 12' before 9 o'clock; 12 hrs. 24'. 1- IfTsBrH 2. (a) 256 shots. (f3) 18-2634375 fr. 3. Edge = 4 ft.; diagonal = 6-928. ..ft. ; area of face = 16 sq. ft. 4. 1125guineasi 5. 89^ days. 6. £3112. 18s. 4ti. 7. £420. 8. 66 planks ; cost = £20. 3*. 4d. 9. 7-9 inches nearly. 10. 3'. 11. 1st 12 o'clock ; 2nd 10^}' and 43xV past 11 o'clock; 3rd 27^' past 11 o'clock. 12. Share left = ^th = -*230769 ; its value = £29-239423076. XII. 1. (a) 450 men. (/3) 135 days. 2. £263. 1 2*. 8d. percentage = 2 A. 3. 10-000000000009. 4. 11 o'clock p. m. 5. 42 cub. ft. 1512 cub. in. ; ^^ih will be filled. 6. 150 qrs. 7. £3863. 12^. S^d. i^q. 8. He gains £28. 15s. 9. 8ff days. A mows 4|f ac, B mows 5f |- ac. 10. £1002. 19s. 3|d. -2448^. U. 128 lines. 12. 15s. 6d. ANSWERS TO THE MISCELLANEOUS PAPERS, (pp. 299-303.) I. 1. A has \\\, B has lj%, and C has X^x- 2. £1549. 3s. Ad, 3. 408 yds. 4. 8s. 5^rf. -3085. 5. 89 yds. 6. ^3; '425. 7. £3. 19s. 3|rf. 8. 17mo.; £474. Ibs.Qld.l^q. 9. £17. 18s. 1^ 10. 3 : 13. 11. £5, 10s. 12, £6-500; 97^. II. I. 7564; 23-919. 3. £1 104. 4fl. 3 c. 1| m. 4. £1760. 5. £7. 9*. lli^c/. 6. 5s. 11^^. t<7. 7. 5 tons, 13cwt., 4oz. 8. 10 ponies. 9. 21 f^ days. 10. £3. 6s. M. ; £3. 15s. ; 8s. Ad. per cent. II. 5 ft. 7^ in. 12. £10666. 13s, 4cf. 13. £1203.155. III. 1. (I) 26^ (2) ly^T". 2. £5. Os. 3^^ 3. Share of each son = £4000, share of eldest daughter = £3000, share o! younger daughters = £2700. 4. £4-08 per cent. 5. (1) 1 yd. ; (2) 99 yds. 6. 6 days. 7. £2. 10s. 8. The second. 9. 65 per cent. 10. 74sq. yds., 2sq. ft., 96sq. in. 11. 24 dozen; 9s. 7d.; 19s. 2ci.; £1. 18s. 4d. 12.£37l|f IV. 1. -031; 1-08 in. 2. -0271. 3. (1) £720, (2) £478. 9fl. 1 c. 4 m. nearly. 4. £647. 10s. 5. 21 ; 20. 6. £196. 13s. 6|r/. 7. 160. 8. £5212|f; £6. 2^. 4jV. 9. £2. 5s. lOrf. by ^, and £2. Is. 8rf. by J?. 10. 850nbs.; £8. 17s.2ic^. H. 10s. 6(f. ^ has to receive £123. 7s. 6d. ; JB, £170. 12s. 6d. ; C, £275. 12s. ej. 12. 3f days. V. I. 600 wks. 2. 4^;4s. 4(i. 3. Each rhan has 15 lb. ; each woman has 7^ lbs. 4. 46980 pieces. 5, £2. 6s. lOi. 6. £41.55. 7. Ikd. f^q. 8. 25^^"^ yds. 9. -083 ; 75-1. 10. £15000. II. In 120 days ; 16' past 2 o'clock and 14' before 2 o'clock. 12. 56^^*. per dozen. 346 ANSWERS (pp. 303-308.) VI. 1. (1)7900000. (2)0. (3)7. 2. 1 m. 5 fur. 20po. 1 yd. 2in.; ISiMffur. 3. £141. 5s. 4. £225; £450; £675. 5. £3. 2s. 6d.; Interest exceeds Discount by lOci. 6. 2 ft. 1 if in, 7. £106. I3s. 4d. 8. 10ft. 9. 25. 10. 8-75ft.; 105in. ll.£1300. Heloses£4. 12. £7 per cwt. VII. I. The 1st article is the cheapest by o^; 2s. 3d. 2. (1) 1. (2) Id. (3) •002668495238089... 4. £2000. 5. ^ does it in 16i^f days ; JB, in 25,^ij days ; and C in 43iV days. 6. 87 Italian lire. 7. £640; £533. 6s. Sd. ; £400; £266. 13s. 4r/. 8. 36 per cent. 9, £82. 9s. Irf. 10. 9009, II. £].0s.9^d.}lq. 12. £9. Os. lOd. VIII. I. Terminating Decimal. 2. -96719... 3. -666... yds. ; 41-52 in. 4. £1389. 35. ; £1680. 5. 5 cub. ft., 579 cub. in. 6. 2.^ per cent. 7. £18630. 8. £1.8s. 6fj. 9. 1079f lire. 10. Price per ton = £1. Is. 6d. ; weight of a sack of coals = 1 cwt. 1^ qr. II. lOhrs. 12.£l0x|-6W per cent, is gained. IX. 1. 900991; -100999; 58045 lbs. 2. (1) jV (2) l| ; 7 ft. 8f in. 3. (1) 4^^; 5-419. (2) -09. (3) -50505; 25000; 45. 4. 9 men. 5. 245 guineas, 189 guineas, 54 guineas. 6. £264. 12s. 7. 5| days. 8. 40yrs. 9. £298. 9 fl. 2c. 9-4 m. 10. £29. 9s. 9lrf. ^6^7- 11. 7^ mo. 12. 7 ft. 51 in. nearly. 13. £1898. 8s. 9c/. X. 1. 41-53... feet. 2. B has to pay A £\. 10s. \d, 3. 54m. ; 3m. 4. 4.1 per cent. 5. 5d. 6.£2|. 7. 2:7. 8. 10 points ii; 70. 9. £416. 14s. ll^(i. 10, 40:41. 11. £6666. 13s. 4cZ. ; £83iV 12. £-242914ftf. XI. 1. •0434027* sec. 2. Each junior partner receives £150; second, £750 ; senior, £900. 3. £488. 5s. 61^. 4. £18^§f|t- per cent. 5. 77|; £1542. 17s. If J. 6. 5 per cent. 7. 3s. lO^rf. t\Wo1J7- 8. £42. 12s. 3iV. ; £9. 13s.2tV- 1<^*- 2005 in. 11. £32000. r2. £112. 10s. The four- oared boat. A^s^SWEllS (pp. 309-314.) 347 XIL 1. -008. 2. 10917437 nearly. 3. '000038... 4. 604 horses. 5. £14. 2fl. 7 c. 5m. nearly. 6. He would increase his income. 7. The first. 8. 5jV ; 54xV; 21iV; 38f/ past 7 o'clock. 9. A does the work in 6 hours ; B, in 12 hours ; and C in 9 hours. 10. £15. 10*-. 3(i. 11. £438. 12. The circuitous exchange XIII. 1. Sd. 2. 3^ yrs. 3. He loses £14. 10s. Sfi. 4. 14 minutes. 5. £394. Os. 3f/. 6. 114 gulden. 7. 38| days. 8. 5^|i}grs. 9. 3 ft. 10 in. 10. 7i per cent, 11. £24. 7s. 4J. 12. U yds. XIV. 1. 34-002; 83100; -000831. 2. 8s. 3. £23. 2s. 9^24^- 4. £107. 66-. 8r/. 5. 10 gulden. 6. £92. 5fl. " 7. £162; £324; £405; £486; £648. 8. 18 days. 9. 55.^. 2-12. 4 ft. 4 in. 10. His income must be £150. 15s. 11. 122 florins. 12. £97826^^77 ; he gains in income £459^§. XV. 1. 48 days. 2. £13. 6s.-8(i.; £4||f-. 3. 641b., 12 oz. 4. Hewillgain. 5. 672 oz. of gold, and 64 oz. of alloy. 6. 176785 tons, 14 cwt., 32 lbs. 7. £ltf. 8. 31 cub. chains, 255875 cub. links. 9. £66. 13.. 7if(i. 10. £1896. 12s. 6(f. 11. 4^ ft. 12. £27691. 13s. 4d. ; 5 per cent. " XVI. 1. £18. 3s. 9jV. 2. £37. 7 fl. 8 c. l|- m. ; oV q. 3. 3 dwts. ig^^-o^grs. 4. 73iJjin. 5.211140. 6. 6A hrs. 7. 34 lbs., 4 oz., 17 dwts., 19-5 grs. 8. 5 hrs. 10'. 20|t". 9. -0000042921... in. 10. £2031. 5s. 11. He gains £5. Us. l^d. per cent. 12. 77 shares ; £46. 4s. XVII. I. 5iV past 1 o'clock. 2. £50. 6s. 3J. 3. 49iV' past 3 o'clock. 4. |. 5. 1-4375 miles. 7. 61-023377953 cub. in. 8. 8 days. 9. £125. 18s. 6£i. 10. 13-1408... per cent. ; 8000000. I I. Int. in 1st case : Int. in 2nd case :: 6800 : 7221. 12. 48 books. CAMBKIDGE: PRINTED BY C. J. CLAY, M.A, AT THE TTNTVERSITY FREfS. BY THE SAME AUTHOR. / A Key to the Arithmetic for Schools. Ninth Edition. Crown 8vo, cloth, 8s. 6d. Arithmetic and Algebra in their Principles and Application : -with numerous systematically arranged Examples, taken from tha Cambridge Examination Papers. Twelfth Edition. Crown Svo, cloth, I OS. 6d. Exercises in Arithmetic. Crown 8vo, 25.; or, with Answers, IS. 6d. Also sold separately, in Two Parts, 15. each; Answers, 6c?. A Shilling Book of Arithmetic for National and Elementary Schools. iSmo. Part I., containing the First Four Eules, in 3 2 pages, sewed in cloth covers, price 2d. This Part contains all that is required of Standards I. and II. in the Government Examination. Part II., containing the Compound Rules, Bills of Parcels, and Practice, in 48 pages, sewed in cloth covers, price 3c?. This Part contains all that is required of Standards III., IV. and V. in the Government Examination. Part III., containing Fractions, Decimals, Rule of Three, The Metric System, &c., in 112 pages, sewed in cloth covers, price 'jd. The Three Parts, complete in i Vol. with the Answers, i8mo, cloth, price IS. 6d. Answers to the Shilling Book of Arithmetic, 6d. Key, i8rao. 4s. 6d. Examination Papers in Arithmetic. In Four Parts. i8mo. IS. 6d. With Answers, is. gd. Key, i8mo. 4s. 6d. The School Class-Book of Arithmetic. Parts I. and II., i8mo, limp cloth, price lod. each. Part III. is. ; or Three Parts in I Vol. price 3s. i8mo, cloth. Key complete, i8mo, 6s. 6d.', or Three Parts, 2S. 6d. each. The Metric System of Arithmetic, its Principles and Appli- cation: with numerous Examples, Written expressly for Standard V. in National Schools. Fourth Edition. iSrao, cloth sewed, ^d. A Chart of the Metric System, for School Walls. On Eoller, IS. 6d. : on Roller, mounted and varnished, 3s. 6d. Fourth Edition, with a full length Metre Measure, subdivided into Decimetres, Centi- metres and Millimetres. Also a Small Chart on Card. id. Easy Lessons in Arithmetic combining Exercises in Reading, Writing, Spelling, and Dictation. Part I. Crown 8vo, gd. The Metric Arithmetic. [Shortly, MA CM ILL AN AND CO. LONDON. WORKS BY REV. BARNARD SMITH, M.A. ARITHMETIC AND ALGEBRA, in their Prin- ciples and Application, with numerous systematically- arranged Examples, taken from the Cambridge Exami- nation Papers, with especial reference to the Ordinary Examination for B.A. Degree. Twelfth Edition. Cro\\Ti 8vo. cloth, los. 6d. ARITHMETIC FOR THE USE OF SCHOOLS. New Edition. Crown 8vo. cloth, 4s, 6d. Answers to all the Questions. KEY TO THE ARITHMETIC, containing Solutions to all the Questions in the latest Edition. Eighth Edition. Crown 8vo. cloth, 8s. 6d. EXERCISES IN ARITHMETIC. Crown 8vo. New Edition, 25. Or with Answers, is. 6d. Also sold separately in Two Parts, is. each. Answers, 6d. ONE HUNDRED EXAMINATION PAPERS IN ARITHMETIC. In Four Parts. iSmo. cloth, is. 6cZ. With Answers, is. gd. KEY TO EXAMINATION PAPERS. iSmo. cloth, 4s. 6d. SCHOOL CLASS BOOK OF ARITHMETIC. Part I. To the end of Compound Division. i8mo. limp cloth, lorf.— Part II. Fractions, Decimals, Extraction of Square and Cube Root. lod. — Part ill. Rule of Three, Interest, &c. is. The Three Parts complete, in i Vol. i8mo. cloth, 35. WORKS BY REV. BARNARD SMITH, M.A. continued. KEY TO CLASS BOOK OF ARITHMETIC. Complete, i8mo. cloth, 6s. 6d.; or separately, Parts I. II. and HI. 23. 6d. each. A SHILLING BOOK OF ARITHMETIC FOR NATIONAL AND ELEMENTARY SCHOOLS. i8mo. cloth. Also in Three Parts. — Part I. containing the First Four Rules, 2d. — Part II. containing the Compound Rules, Bills of Parcels, and Practice, ^d. — Part III. containing Fractions, Decimals, Rule of Three, the Metric System, &c. •jd. ANSWERS, 6d. With Answers, complete in I Vol i8mo. cloth, is. 6d. KEY, 4s. 6d. THE METRIC SYSTEM OF ARITHMETIC, its Principles and Application, with numerous Examples. Written expressly for Standard V. in National Schools. Fourth Edition. iSmo. cloth, sewed, ^d. A CHART OF THE METRIC SYSTEM for School Walls, on KoUer, is. 6d. ; on Roller, mounted and varnished, Fourth Edition, 3s. 6d. A small Chart on Card, id. EASY LESSONS IN ARITHMETIC, combining Lessons in Reading, Writing, Spelling, and Dictation. Part I. for Standaid I. in National Schools. iSmo. cloth sewed, ^d. DIAGRAMS FOR SCHOOL WALLS in Prepa- ration. macmilla:^ >AS^"-CO. london. ••/ " y i^^t ^^q i ~ lltf -xi '• "% /\- \ s ^ ■'^^.: THIS BOOK IS DUB ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS W.Ul. BE ASSESSED ^Oll^^'i^'^J^ ^^^^^ -rule: nndK ON THE DATE DUE. THE PENAUIt w^Ll Increase TO so cents on the fourth olv ^ND TO%..0O ON THE SEVENTH DAY OVERDUE. '^^'^.^ LD 21-100m-8,'34 iluP'-. W I /4ZV 6 A '6 z