nfn 4-' }7i %ii>. ^itJ 07 t IN MEMORIAM FLORIAN CAJORl RATIONAL ARITHMETIC, IN WHICH THE SCIENCE IS FULLY DEVELOPED, THE ART C L E A R T. Y T^ X" P I A I \ E D , AND BOTH COMBINED IN NUMEROUS ILLUSTRATIONS ; ADAPTED TO LEABNER4 OF EVERY CAPACITY. THL WHOLE ENFORCED BY A GREAT VARIETY OF INTERESTING AND PRACTICAL PROBLEMS. TO WHICH IS APPENDED, * Jk EISY, CONTAINING THE ANSWERS TO THE PROBLEMS. BY J. S. RUSSELL, TXACHKR OP MATHBMATICS kit TRB LOWBLL HIOB SCHOOL. SECOND E4)ITI0N. LO WELL: PUBLISHED BY THOMAS BILLINGS BOSTON B. B. MUSSEY. 1S47 Entered according to Act of Congress, in the yesr 1846, by J. S. RUSSELL, In the Clerk's Office of the District Court of Massachusetu. <> Stereotyped bj GEORGE A. CURTIS; MBW MH^LAMV TITS AXC TBAKOTT7S VOUWMIV w PREFACE. Every public school may be divided, in respect to the stad^ of Arithmetic, into three classes. The first and smallest class either possess by nature, or have happily acquired, a taste, and, conse- quently, a talent for the study. For them there is no imperious necessity of adding another to the numerous treatises already m use ; for, although they will meet with much difficulty, through mdefinite and confused modes of expression and incomplete demonstrations, in arriving at the philosophy of the subject, yet, in spite of these obsta- cles, they will eventually comprehend the important principles of Arithmetic, and, what is remarkable, adopt the same modes of expression which have so much opposed their own progroBS. Tq this class the Rational Arithmetic, though not indispensable, will be of very essential service. Had all learners been of this class, however, the author would have been spared the labor and expense he has devoted to this work. But there is a class of medium ability, including about one half, who may be saved incalculable labor and vexation, by using this book, while pursuing this difficult study. It is expected, however, that the third class will most truly appre- ciate this work. They include about one third, and consist of those unfortunate scholars whose minds act too slowly for the patience of teachers, and are too obtuse to derive much advantage from the text- books so ill-adapted to their wants. It is for these two classes, the last in particular, that the Rational Arithmetic is prepared ; to their wants it is thought to be well adapted ; and it is expected, that they will hereafter assume a more just relative standing with their schoolmates ; not waiting, as hereto- fore, for the results of business life ib prove them possessed of minds, less active indeed, yet not inferior in strength and capacity of im- provement. The author knows no other written Arithmetic that is adapted to learners ; they seem to him rather books of reference for those who already understand the subject, and are able to perceive the princi- IV PREFACE. pies Without exp.anation. Some, indeed, have attempted to meet the wants of learners, by introducing the several subjects by a page or two of puerile questions which arc seldom noticed eitherby teachers or scholars. Others found the principles upon the imagi- nary answers to be ffiven to such questions by those acknowledged to be ignorant. Such must be a very uncertain foundatioiif especially so when, as sometimes happens, theee l^-^'^-^" v^estioiie are so misformed as to lead astray. Instanci* the : * In 11, how much more does the 1 in the lens' place 8tiv,. . . :aan the 1 in the unit's place? In 880, how much more does the 8 in the hundreds' place stand for than the 8 in the tens' place 1 It is just so in all cases ; therefore, A figure at the left of another stands for ten times as much as it would m the place of that other figure.*^ The simple learner will, probably, understand these questions literally, and give for answers, so far as he is able, 9 more, 720 more, &c., between which and the principle purporting to be derived from them, there is no direct connection. Had these questions been thus, How many times as much as 1 insiea'* " .w much more than 1 ^ they would have i 1- ligent mind directly to the principle. In presence of the u;^nur to correct false answers, to sum up and enforce the cooclueion, such questions properly asked are well enough ; but otherwise, they are extremely vexatious and discouraging, and meet echolars will pass over them without adding to their knowledge. In the Rational Aritlnnetic it has been the object to prepare matter for the intelligent study of the learner by himself y that in due time he may, in a well conducted recitation, exhibit with credit and pleasure, both to himself and teacher, his thorough knowledge of the lesson. Such results are far different from ordinary experience. Teachers who have desired to ground their pupils in the principles of the science, while the text-books have failed to afford the necessary instruction, have, by oral instruction, and black-board iUustrations, endeavored, with only partial success, to effect this object, at present, so indispensable. Such instruction, though efficient with tlie more inteUigent and active minds, proves insufficient for a large portion of every school. It is expected that the Rational Arithmetic will come to the relief of such teachers, enabling them with less labor to secure much happier results. The peculiarities of the Rational Arithmetic are : 1st, A philosoph- ical arrangement y and systematic treatment of the several subjects. Multiplication and Division, being only particular cases of Addition and Subtraction, respectively, follow their heads in the natural order, in the fundamental principles ; but in fractions and compound num- bers, from the greater convenience, they resume the common order again. T\\e subjects under the head of Percentage, being applica- tions of the principle of Proportion, very properly follow Proportion. The study of Arithmetic being now so extensive, it is no longer necessary to place Interest nearer the beginning of the book than its PREFACE. T proper place, to insure a knowledge of it. Indeed, all the subjecta are so arranged that each is explained on' principles previously taught. 2d. A complete devehpmeiU of the fundamental principles. Nume- ration, in particular, both integral and fractional, the foundation of the whole superstnirture, has received especial attention. 3d. A full (! L and thorough explanation of the various applications of i nontal principles. 4th. A < jard to the abridgment of labor, by viewing num- bers ihro! liictors, and relations, canceling common factors when both m. -i and division are involved in the same pro- cess ; and alwa ig-'upon fractions in a manner to secure the simplest terms i ...^. 5th. The giving of a reason for everything stated, and in such style that the repetition of the language will induce in the learners the understanding of the reason which it embodies. 6th. Numerous references to other parts of the book for informa- tion bearing upon the subject in hand. 7th. An exclusion of all such indefinite expressions as ** 5 times greater," ** seven times too large," "seven times too small," ** in- creases it ten times," ** 5 times too great," " 100 times larger or smaller ;" and all such provoking expressions as ** it is obvious," " it is plain," ** evidently," &c., whose office is only to occOpy the place of an inconvenient reason. These peculiar excellences, it b thought, warrant the title assumed for the book. It is recommended to teachers, although the author knows no writ- ten arithmetic so easily understood, that the younger pupils, pre- viously to taking up this book, shall have well studied Colbunrs First Lessons, or some other intellectual arithmetic ; but when it is taken up, that they accommodate their speed to thoroughness ; that they take special notice of the numerous references to other parts of the book, where their memories may be refreshed with necessary information upon the present subject. Teachers will, of course, as far as circumstances admit, classify their pupils, assign lessons, and hear recitations in this, as in other studies. Although each is left to his own experience and tact in conducting recitations, yet we may urge the importance of securing in some way a thorough analysis of everything^ the giving of a reason for each step in the solution of problems, showing its bearing upon, or tendency to war Is the final result. In the author's experience, it has proved well to require the pupils to bring to the recitation, not only the results, but the written process of their work ; also to exhibit their skill in the solution of problents upon a black-board sufficiently ample \ > accommodate the whole class. The problems may be those of the ordinary lesson, or such as may be suggested on the occasion ; and the recitation should be such as shall exhibit the scholar's knowledge of the principles involved in the process. The impossibility of preventing the access of the scholars to th 1* 1 PREFACE. " Key published for the use of teachers only^^* the immenae injury done to the moral sense by the futile attempt, the securing of greater faithfulness on the part of teachers, and, on the other hand, the accommodation of the better scholars, who dislike to have the answers obtruded upon their notice before they shall have given the problem a fair trial, with other reasons, have induced ih'^ author to ap|)end the Key to the Arithmetic, where it may be cm v, and innocently accessible to all. Hut should any prefer li ;cn Process and Proof of Division, Short Process of Division , Long Process of Division General Exercises, FRACTIONS Origin and Mode of writing Fractioi Definitions, Reading of Fractions, Expression of Division, Finding the Whole from a Tail, . Finding a Part from the Whole, ]\I()des of considering and ^ " ' ^ ; < ...n^. Expression, Definitions, :i n o| Fraction.*-. Reduction of Mixed Nuin 'T Fractions, . Multiplication of Fractioi umbers, . . Division of Fractions by 1 : is, . , . Dividing by the Factors of ihc Divisor, Reduction of Fractions to lower terms. ' - '^' -rs, H'lor, ^ V , ,...iiple and Denominator Addition and Subtraction of Fractions, Multiplying by Fractions, Dividing by Fractions, Review of Multiplication of Fractions by Fractions, . Review of Division of Fractions by Fractions, . . . DECIMAL FRACTIONS. Similarity of Decimal Fractions and Integral Numbers, Local Value of r' ''""'- Reading of Dt' Writing of lk\ Federal Money expressed by Decimal Number- Reduction of Federal Money, Addition and Subtraction of Decimals, Multiplication of Decimals, Reduction of Common Fractions to Decimals, . . . Dividing by Units of the higher orders, Infinite Decimals, Division of Decimol- COMrOU.ND NU3IBERS. Definitions, Tables, Reduction of Compound Numbers, Addition of Compound Numbers, Subtraction of Compound Numbers, Multiplication of Compound Numbers, Division of Compound Numbers, BCnOH. 70 to 76 77 78 79 80 81 82^85 83 84 86to8S 89 90 919:; 100109 110116 117119 120 130 123124 128130 131 140 141 143 144 150 151 157 158 159 160161 162 163 164 165 166 168 169 170172 173 - 174 175177 178179 180 182 183 187 188191 192 193206 207233 234 236 237 239 240 24 242 243 CONTENTS. 9 CBcnoif. Using of Numbers variously expressed, 244 lo 245 Finding the Difference of Time between Dates, .... 246 24B Reduction of Compound Numbers for Multiplication, . 248 249 Mensuration of Surfaces and Solids, 250 251 Abridged Solutions of Problems, 252 253 Reduction of Currencies, 254 255 Practice, or the Use of Aliquot Parts, 256 263 PROPORTION. Ratio, 264-265 TVT'iitiniving by Ratios, 266 271 g by Inverse Ratios, 272 275 ] 276278 ] (jportion, 279 280 c Proportion, 281 284 Conjoined Proportion, 285 286 Barter, 287 290 Fellowship, 291 292 Compound Fellowship, 293 294 PERCENTAGE. Percentage, 295 297 Commission, 298 300 Stocks, 301 302 Insurance, 303 304 Assessment of Taxes, 305 Duties, 306 309 Interest, 310 313 Interest on Partial Payments, 314 316 Banking, 317 320 ComJ)ound Interest, 321 323 Compound Interest on Partial Payments, 324 325 Problem to find the Time, 326 328 Problem to find the Per Cent., 329 331 Problem to find the Rate Per Cent., 332 334 Problem to find the Principal, 335 337 Problem to find the Present Worth, 338 340 Problem to find the Discount, 341 343 Problem to find the Face of a Discounted Note, . . . . 344 346 Problem for the Equation of Payments, 347 349 ALLIGATION. Problem to find the Average of Ingredients, 350 352 Problem to find the Quantities of Ingredients, .... 353 356 Problem to mix Ingredients partially limited, .... 357 359 Problem to mix a Limited Compound,* 360 362 POWERS AND ROOTS. Definitions and Illustrations, 363 364 Extraction of the Square Root, 365 372 Extraction of the Cube Root, 373 381 10 CONTENTS. SERIES. r in Scries by Difference, > find either Extreme and the Sum, ) find the Common Difference and Sum, ) find the Number of Temw and Sum, i iis in Series by Quotient, Problem to form Series, Problem to find either Extreme and Po-wcr of the Ratio Problem to find the Sum, Infinite Series, . . . Com; ' 'forest by Senes, Com ount by Series^ Aniiu...; ..ned, .... Annuities at Simple Interest, . Annuities at Compound Interest MENSURATION. Definitions, Mcnsnmtion of the Parallelogram and Triangle, . . !on of the r ion of the I ion of the Cyiimier, ion of the Pyramid, Cone, and Wedge, . . ion of the Frustum of the Pyramid and Cone, on of the Sphere, < _ _ of Casks, ... REVIEW. Review of Fractions, Review of Compound Numbers, . . Review of Proportion, Review of Percentage, Review of Alligation, Review of Powers and Roots, . Review of Series, Review of Mensuration, Curious Problems, 382 383 to 385 386 388 389 391 392 393 396 397 400 401 403 404 406 407 409 410 412 413 414 419 420 428 429 430 433 433 435 436 437 438 439 440 441 442 443 444 445 446447 448 449 450 451 452 453 454 455 456 KEY. Containing the Answers to all the Problems. % ARITHMETIC. I. NUMERATION. ! Arithmetic Defined. Arithmetic is the science of numbers, and the art of com- puting by them. As a science, arithmetic explains the nature and properties of numbers ; and demonstrates the principles and rules for the practice of the art. As an art, arithmetic explains the methods of working by numbers for the solution of numerical problems. 3, Formation of Numbers. A single thing of aoy kind is called a unit, or One. The larger numbers are formed by the successive addition of units. Thus,' if to one, another unit of the same kind be added, the collection forms the number. Two. The collection of two and one forms the number, Three. The collection of three and one forms the number. Four. The collection of four and one forms the number. Five. The collection of five and one forms the number. Six. The collection of six and one forms the number. Seven The collection of seven and one forms the number. Eight. The collection of eight and one forms the number. Nine. In like manner, the addition of one unit to any number, forms the next larger number. 3. Arabic Figures. Among the \^arious methods of expressing numbers, the Arabic is superior ; and is now in general use. According to this method, all numbers can be expressed by different combinations of one, or more, of ten figures. The figures are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine figures are also called digits. And each digit, expressing one of the first nine numbers, has the samp name as the number which it expresses. 12 ARITHMETIC. Thus: One is expressed by this figure, 1, called One. ^ Two is expressed by this figure, 2, called Two. J Three is expressed by this figure, 3, called Three. Four is expressed by this figure, 4, called Four. Five is expressed by this figure, 5, called Five. Six is expressed by this figure, 6, called Six Seven is expressed by this figure, 7, called ^ Eight is expressed by this figure, 8, called L.^.... Nine is expressed by this figure, 9, called Nine. The other figure, 0, called Cipher, unlike the digits, does not express a number, nor have any value; but yet, as we shall see, it is not without its use. 4. Expression of Tens, or Units of the Second Order. There is no appropriate figure to express the next number, called ten, or any of the larger numbers ; but these same digits are made to express other numbers by occupying dif- ferent places in relation to each other. ^^ hen they stand alone, or in the Jirst place, each expresses a certain number of units of the first order. But ten units of this order are considered collectively as forming one unit of the secoTid order ; and the digits are made to express units of the second order, called tens, by occupying the second place from the right hand. Thus: 10 is one ten, called Ten. 20 is two tens, called Twenty. 30 is three tens, called Thirty. 40 is four tens, called Forty. 50 is five te^is, called Fifty. 60 is six te/is, called Sixty. 70 is seven tens, called Seventy. SO is eight tens, called Eighty. 90 is nine te?is, called Ninety. Here the digits express ten times- as much, or numbers ten limes as large, as when they stand in the first place, or alone, because they occupy the second place, and not because there is any value in the cipher. The cipher merely occupies the fiist place, in order that there may be a second place for the dif^it to occupy. So always, the cipher is used to occupy places where nothing of value is needed ; but which must be occupied, in order that the digits required for the expression of the number, may stand in tli^ir proper plsces. NUBCERATION. 13 S* Expression OF Numbers from Ten to One Hundred. The numbers between the tens, that is, between ten and twenty, twenty and thirty, &c., are expressed by making every digit, in succession, occupy the first place, together with each digit in the second place. Thus : s one unit of the second order, called Ten. s ten and one, called Eleven. s ten and two, called Twelve. s ten and three, culled Thirteen. 8 ten and four, called Fourteen. s ten and five, called Fifteen. s ten and six, called Sixteen. s ten and seven, called Seventeen. s ten and eight, called Eighteen. s ten and nine, called Nineteen. 8 two tens, or two units of the second order, called Twenty s two tens and one, called Twenty-one. s two tens and two, called Twenty-two. s two tens and three, called Twenty-three. s two tens and four, called Twenty-four. s two tens and five, called Twenty-five. s two tens and six, called Twenty-six. s two tens and seven, called Twenty-seven. s two tens and eight, called Twenty-eight. is two tens and nine, called Twenty-nine. s thr^ tens, or units of the second order, called Thirty. s three tens and one, called Thirty-one. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28^ 29 30 31 32 &c., is three tens and two, called Thirty-two. 41 &c., is four tens and one, called Forty-one. 51 &c., js five lens and one, called Fifty-one. 61 &c., is six tens and one, called Sixty-one. 71 &c., is seven tens and one, called Seventy-one. 81 &c., is eight tens and one, called Eighty-one. 91 &c., is nine tens and one, called Ninety-one. 99 16 nine tens and nine, called Ninety-nine. 6. Expression of Hundreds, or Units of the Third Ord^r. Ninety-nine is the largest number that can be expressed by two figures. The next larger number is ten tens, or ten units of the second order, which are considered collectively as forming one unit of the third order, called one hundred. 2 14 ARITHMETIC. This unit is also expressed by the tirst digit; but, it being a unit of the third order, the digit is put in the third place. And the other digits, by occupying the third place, are made to express units of the third oraer, or hundreds. Thus : 100 is One hundred. 600 is Six hundred. 200 is Two hundred. 700 is Seven hundred. 300 is Three hundred. 800 is Eight hundred. 400 is Four hwidred. 900 is Nine hundred. 500 is Five hundred. T. Expression op Numbers from One Hundred to Onb Thousand. The numbers between the hundreds are expressed, by making all the numbers less than one hundred, in succession, occupy their own places at the right of each digit in the third place. Thus: 101 is one hundred and one. 210 is two hundred and ten. 311 is three hundred and eleven. 425 is four hundred and twenty-five. 543 is ^\e hundred and forty-three. 608 is six hundred and eight. 717 is seven hundred and seventeen. 876 is eight hundred and seventy-six. 999 is nine hundred and ninety-nine. Expression of Thousands, or Units of the Fourth Order. Nine hundred and ninety-nine is the largest number that can be expressed by three figures. The next larger number is ten hundreds, or ten units of the third order, which are considered collectively as forming one unit of the fourth order, called 07ie thoitsand. To express thousands, or units of the fourth order^ the digits are put in \\ie fourth place, 9. Expression of Numbers from One Thousand to Ten Thousand. Any number between the thousands is expressed by using such digits as are needed in their proper places at the right of the thousands. Thus : NUMERATION. ij 1000 is one thousand. 2001 is two thousand and one. . .'* 3020 is three thousand and twenty. 4500 is four thousand and five hundred. 5055 is five thousand and fifty-five. 6107 is six thousand one hundred and seven. 7819 is seven thousand eight hundred and ninetecii. SOU is eight thousand and eleven. 9999 is nine thousand nine hundred and ninety-nine. 10. Expression of Ten-Thousands and Units of other Orders. Ten units of the fourth ardery form one unit of the fifth order, called a ten-thoiLsand. And the ten-thousands must occupy the fifth place. In the same manner, higher orders of units are formed, to an unlimited 'extent ; ten units of any order forming one unit of the next higher order, to be expressed in the next higher place ; while the lower places are used for the expression of units of lower orders. Whence it follows that one unit of any order equals ten units of the next lower order; this law prevailing, even below units of the first order, to an unlimited extent, as will be shown, (183) Hence the law of the local value of figures is, that any digit, by each removal to the next higher place, is made to express ten times as much, and by each removal to the next lower place, one tenth as much as it would before such removal. N. B. The term unit means a unit of the first, or lowest order, unless otherwise specified. 11. T^BLE exhibiting THE FORMATION, NaME, AND EXPRES- SION OF ONE Unit of each of the first Ten Orders. A single thing of any kind forms one Unit, 1. Ten units of the same kind form one Ten, 10. Ten tens form one Hundred, 100. Ten hundreds form one Thousand, 1000. Ten thousands form*one Ten-thousand, 10000. Ten teu-thousands form one Hundred-thous. 100000. Ten hundred-thousands form one Million, 1000000. Ten millions form one Ten-million, 10000000. Ten ten-millions form one Hundred-million, 100000000. Ten hundred-millions form one Billion. 1000000000. 16 ARITHMETIC. o S . ly lirst hy the 2 units, ihcrj l)y iIh^ 2 tens, or 20, nnd then by the 2 hundrrds, or 200, ar- ^nM ranging the j>rodurts to^'etlier, so that the units of thv. same orders may stand in the same roh/nins. Multi- ply hy the 2 uiiitH, as usual. The factors of 20 beintf 2 and 10, nudtiply hy the 2, and make this product 10 1dA7d ^""^''** ^^ larj^e, (99,) l>y vvrilint: it one degree to the ^^^^ left, (4.) The factors of 200 heing 2 and 100, multi- ply by 2, and make this product 100 timrs as large, by writing it two degrees to the left, (SM.) 1'l>'" ^'nn of \\wne products will be the answer recjuired. 2. Multiply 20003 by 1007. '>00(n ^^^^^^ ^^^'^^' '^'^* multiplicand 7 times, then 1 thou- ^1007 ^'^'^^^ times. To nmltiply by the 1000, write onc(^ the multiplicund, three degrees to the left, 140021 (JM.) The three ciphers neetj not Ik? armexed ; 20003 for, without them, each figun^ of this product will om^QH^T ^^^' ^^^ ^^*^* column of units of its own order, and -iU14.lU^l ^i^j^rofore will be added in the right plare. 4S MoDKL OK A RkCITATION. How nmch is 30r)08 times 403070 ? /in*^n7n l^^'r(^ the nndtiplicand is to be taken S times, on^nfi 500 times, and 30000 times. Multiply by the .5U0UO Q rp^^^^ p^^,^^^^^ ^^^^3^ l^^,j^^ ^ ^jj^, j^^^ ^^jj^ 3224/500 ^^P'y ^^y ^' '^^''^ make this j)ro(luct 100 limes 201/)3f50 ^^ I'vrge, by writing it iip two degrees. The 1209210 factors of 30000 })eing 3 and lOOOO, multiply by 3, and make this prod tic t 10000 times as I2296a59660 large, hy writing it up four degrees. Th MULTIPLICATION. U Bum of these partial products, thus arranged, will be Uie pro- duct lequired. ^-l. OnSKIlVATION. Obskrve, that in these operations (49, 43) the midtipli' Zand is multiplied by each digit in the multiplier^ that the first figure in each partial product is of the same denomination as the mulliplyiiig figure y and that the sum of Oie partial pro- ducts is the product required. 4A. Exercises in Multiplication. In like manner, solve and explain the following problems, 1. A man travelled 26 days, at the rate of 47 miles a day. How fnr did he travel ? 2. If a chaise wheel turn round 346 times in 1 mile, how many times will it revolve in the 25 miles from Boston to Lowell ( 3. How much money would be required to pay 37 men 7/5 (lolhirs apiece ? 4. Wlijit must I pay for 29 fat oxen, at 43 dollars apiece ? 5. What will 97 tons of iron come to, at 57 dollars a ton ? 6. If a vessel sail 158 miles a day, how far would it sail ill [hv. month of April ? 7. If 786 yards of^cloth are made, daily, in a factory which luris 313 days a year, what is made annually in that fac- tory? , a. How much wheat can be raised on 95 acres, at 38 bushels an acre ? 9. Multiply 1728 by 144. 10. How much is 4004 times 999 ? 11. What is the product of 6075 and 67 ? 12. How much is 160012 X 333? 13. Multiply 1H36 hy 1010. 14. Mnliiply 1111 by 2222. 15. Multiply 2222 by 1111. 16. Multiply 3000024 by 309. 17. Multi])ly 309 by 3000024. 40* Model of a Recitation. What is the product of 32000 and 2300 ? 34 ARITHMETIC. 32000 '^^^ factors of 2300 being 23 and 100, multiply 2S00 ^y ^^^ placing it under the digits of the multiph- cand, and multiplying without regard to the ci- q^ phers on the right. This gives 736. But, since ^A 23 times 32 units of any order will be 736 units of the sa7?i order, as surely as 23 times 32 things 7*^600000 ^^ ^^y ^^^^^ ^^'^^^ ^^'^ ^^^ things of the same kind ; 23 times 32 thousands will be 736 thou- sands. Therefore annex three ciphers, (845) that it may have the thousands' place ; then annex two ciphers more, (30^) to multiply by the other factor in the multiplier ; which gives the product required. 47. Observation. Observe, that, by this process, (4O5) as many ciphers wiU be annexed to the product of the digits as there are on the right of both factors. 48. Exercises in Multiplying, when the Factors ex- press Units of the higher Orders. In like manner y' solve and explain the following problems. 1. How far is it from Boston to Liverpool, if a vessel sail from Boston at the rate of 150 miles a day, and arrive at Liverpool in 20 days ? 2. How far from the earth to the sun, if it take light 480 ^seconds to come from the sun, at 200000 miles a second ? 3. What is the capital of Boston Bank, there being 12000 shares, at 50 dollars a share ? 4. What is the capital of Massachusetts Bank, there being 8200 shares, at 250 dollars a share ? 5. State Bank has 30000 shares, at 60 dollars each. What is its capital ? 6. If Massachusetts' house of representatives has 500 mem- bers, and a session lasts 90 days ; how much money would it take to pay 2 dollars a day to each member ? 7. How many weekly newspapers will it require to furnish 30000 subscribers one year ? S. What would be the cost of a railroad, 40 miles in length, at 40000 dollars a mile ? 9. Multiply 740 by 6050. 10. Multiply 6050 by 740. MULTIPLICATION. 3S low many are 3400 times 390 ? 12. How many are 390 times 3400 ? 13. What is the product of 140 multiplied by 140 ? 14. Multiply 1600 by itself. 15. 55500 X 4400 is how much ? 16. 1910 X 170 are how many ? 17. If the multiplicand be 160000, and the multiplier 2400, what will be the product ? 18. 121212x8080 = ? 49. General Exercises in Addition and Multiplication. 1. How many months was Andrew Jackson president? 2. How many months was John Quincy Adams president ? 3. How many pounds of pork on 150 wagons, each loaded with 6 barrels, with 200 pounds in a barrel ? 4. If a house have 20 windows, of 24 panes each, how many panes in all the windows ? 5. W hat number is 9000 times 165 ? 6. What number contains 144 twelve times ? ' 7. What number contains one thousand and fifteen 607 times ? 8. What would be the sum of 457 set down ten thousand times, and added up ? 9. What is the cost of a road 40 miles long, of which one half cost 1750 dollars a mile, and the other half, 1800 dollars a mile ? 10. If a quantity of provisions would last 500 men 30 days, how long would it last 1 man ? 11. How many men would consume in 1 day what would last 500 men 30 days ? 12. If a bushel of wheat afford 70 ten-cent loaves, how many cent loaves may be obtained from it ? 13. How many yards of cloth, 1 quarter wide, are equal to 27 yards 5 quarters wide ? 14. How long would it take a man, working 1 hour a day, to do what he could in 26 days, working 12 hours a day ? 15. If a boy attend school constantly 3 terms of 12 weeks, and 1 term of 1 1 weeks ; how many hours is he in school, at 33 hours a week ? 16. How many strokes will the city clock strike in the month of June ? 17. If it take 594 bricks to pave I rod of side-walk, how many would it take to pave a walk a mile long ? 36 ARITHMETIC. 18. What are a man's annual expenses, who pays 3 dollars a week for board, 6 dollars a month for clothes, 10 dollars a quarter for travelling expenses, 1 dollar a week for benevolent purposes, and for other items 75 dollars ? 19. What is a man's income, who receives a salary of 15 dollars a week, and 10 dollars a month interest money ? 20. What is the value of a drove of cattle, consisting of 12 oxen at 55 dollars apiece, 15 cows at 30 dollars apiece, 18 heifers at 16 dollars apiece, and 14 yearlings at 10 dollars apiece ? , 21. What is the amount of the following bill ? Boston, April 25, 1846. Mr. John Merchant, Bought of Charles Wholesale, 27 yards of Black Broadcloth, at $6 a yard, 25 " Blue " u 7 u 18 ** Drab Cassimere, " 3 " 24 Vest Patterns, *' 2 a pattern. Received payment, Charles Wholesale. 22. What is the foot of the following bill ? Hanks, Harris & Co. ^^^^^' ^P^^^ ^^^ ^^^- Bought of Burt & Townsend, 1200 pairs Boys' Shoes, < $ 1 per pair, 400 " Men's " 2 " . . 600 * " Boots, ( 3 " . .* 23. What is the foot of the following accbunt ? Mr. Isaac Speculator, 1846. To Jonathan Farmer, Dr. Jan. 31. To 17 Cords Wood, (8) $7 per cord, Aug. 1. " 9 Tons Hay, (b 15 " ton, Oct. 12. ' 10 Loads Potatoes, (S> 8 '' load, " 15. ' 18 Barrels Apples, < 2 ' barrel. 24. How many scholars can a school-room accommodate, in which are 4 divisions of seats, 11 rows of seats in each division, and 6 seats in a row ? MULTIPLICATION. 37 25. How many are 4 X H X 6 ? 26. How many seats in a church, in which the body pews are in 4 roVs of 18 pews each, the wall pews in 2 rows of 24 pews each, and the gallery pews in 12 rows of 4 pews each, there being 6 seats in each pew ? 27. How many shingles will cover the roof of a house, each of the two sides being 32 feet long and 16 feet wide ; if it take 3 shingles to extend a foot in each direction ? 28. What is the product of 32 X 3 X 16 X 3 X 2 ? 29. If the earth move in its orbit 68000 miles an hour, how far does it move in 24 hours ? 30. How far in its orbit does the earth move in the month of February ? 31. How far does the earth move in the 4 months which have 30 days each ? 32. How far does the earth move in the 7 months which have 31 days each ? 33. How many miles does the earth move in a year, as shown in the last three problems ? 34. If the moon is 240000 miles distant, and the sun is 400 times as far ofl', what is the distance of the sun ? 35. What number is that whose factors are 3, 5, 7 ? 36. What is the product of the first ten prime numbers ? 37. What sum of money must be divided among 27 men, so that each man may receive 115 dollars ? 38. Two men depart, in opposite directions, from the same place, one at the rate of 27, and the other 31 miles a day. How far are they apart in a week ? 39. Two men depart, in the same direction, from the same place ; but one travels 10 miles a day farther than the other. How far apart are they in a week ? 40. The product of two equal factors being called the secona potaer, or square of that repeated factor, what is the second power of 12 ? Ans. 12 X 12 = 144. 41. What is the second power of 15 ? 42. What is the second power of 30 ? 43. What IS the square of 50 ? 44. What is the square of 100 ? 45. The product of three equal factors being called the third power, or cube of that repeated factor, what is the cube of 12 Ans. 12 X 12 X 12 =. 1728. 4 38 ARITHMETIC. 46. What is the cube of lo ? 47. What is the third power of 9 ? 48. What is third power of 25 ? 49. The product of four equal factors being called the fourth power of that repeated factor, what is the fourth power of 3 ? Answer, 3x3x3x3 = 81. 50. What is the fourth power of 5 ? 51. Any number being the first power of itself, what are the first ten powers of 2 ? 52. What are the first ten powers of 10 ? 53. Multiply 144 by the third power of 10. 54. Multiply 18 by the fifth power of 10. ^^, Multiply 500 by the second power of 10. IV. SUBTRACTION. SO. The Principles of Subtraction Illustrated. In Numeration (2) you were taught that the addition of one unit to any number, formed the next larger number. Hence, it follows that taking one unit from any htimber, leaves the next smaller number. In Addition you were taught that two or more numbers, consisting of any number of units, could be united into one larger number, equal to their sum. Hence, it follows that any number can he separated into two or more smaller numbers^ the sum of lohich equals the original number. The father of John and Henr^^ promised to give them 10 cents ; but, as John was the older boy, he should have 7 cents, and Henry might have the remainder of them. Henry, in trying to make his part as many as possible, studied out these curious questions. 1. Hoio many ivill remain, when John has taken 7 from the 10 cents? 2. How many more are the whole 10, than John's 7 cents ? 3. How many less than the whole 10, are John's 7 cents ? 4. Hoio many must he added to John's 7, to make the whole 10 cents ? 6. How many must I take from the whole 10, to leave John's 7 cents ? SUBTRACTION. 39 6. What is the difference between John's part, and the whole 10 cents ? 7. What is tke difference between the whole 10 cents, and John's part ? 8. If 10 cents are separated mto two parts, one of which is 7, what is the other part 1 But he found that, to answer all his questions, he had only to take 7 from 10, and that, in every case, only 3 cents remained for his part. til. Definitions of Terms, and the Sign for Sub- traction. Subtraction is the taking from a number. Minuend is a given number to be diminished hy subtraction. Subtrahend is a given number to be subtracted. By subtraction the minuend is separated into two parts, one of which equals the subtrahend. To ascertain the other part, is the purpose of the opera- tion. This is done by taking the subtrahend from the minuend. The number which is left is the part required, and is called the Remainder. It is the Difference between the minuend and subtrahend. Observe, in the questions above, (SOj) that 10 is the given number to be separated into two parts, and, therefore, is the Minuend ; that 7 is the given part of the minuend, and, there- fore, is the Subtrahend ; that 3 is the other part, or Remainder of the minuend, and, that the two parts of the minuend, 7 -(- 3 = 10, the whole minuend. Observe also, in these questions, the different uses of sub- traction. One horizontal line is the sign for subtraction. It implies that the number which follows the sign, is to be taken from what precedes it, thus : 10 7 = 3, which is read, 10 minus 7 equals 3. Minus is the Latin word for less, and, here, means the same as if you should say 10 less 7, or 7 less than 10 equals 3. 10 is the minuend, 7 the subtrahend ; and 3 is the difference. 52. Subtraction Table. In order to perform subtraction with facility, you will, before attempting further progress, correctly ascertain, and 40 ARITHMETIC. thoroughly commit to memory, the difference between the two numbers of each combination in the following table. 2 2 = 3 2 = 4 2 = 5 2 = 6 2 = 7 2 = 8 2 = 9 2 = 10 2 = 11 2 = 6 6 = 7 6 = 8 6 = 9 6 = 10 6 = 11 6 = 12 6 = 13 6 = 14 6 = 15 6 = 3 3 = 4 3 = 5 3 = 6 3 = 7 3 = 8 3 = 9 3 = 10 3 = 11 3 = 12 3 = 7 7 = 8 7 = 9 7 = 10 7 = 11 7 = 12 7 = 13 7 = 14 7 = 15 7 = 16 7 = 4 4 = 5 4 = 6 4 = 7 4 = 8 4 = 9 4 = 10 4 = 11 4 = 12 4 = 13 4 = 8 8 = 9 8 = 10 8 = 11 8 = 12 8 = 13 8 = 14 8 = 15 8 = 16 8 = 17 8 = 6 !^z=z 6 5 = 7 5 = 8 5 = 9 5 = 10 5 = 11 5 = 12 5 = 13 5 = 14 5 = 9 9 = 10 9 = 11 9 = 12 9 = 13 9 = 14 9 = 15 9 = 16 9 = 17 9 = 18 9 = 53* Model of a Recitation. 1. A man bought a farm for 2325 dollars, and sold it for 2548 dollars. How many dollars did he gain ? He gained the difference between what he gave, and what he received for his farm. Here, 2548 is the minuend, (as the larger of the two given numbers, when there is any difference between them, is always the minuend,) and 2325 is the subtrahend. It will be most convenient to take the units of each order from units of the same order, beginning with the lowest. Therefore, write the sub- trahend under the minuend, placing the units of each order under those of the same order. Take 5 units from 8 units, and 3 units remain, which write in the units' place ; 2 tens from 4 tens, 2 tens remain, which write in the tens' place ; 3 hundreds from 5 hundreds, 2 hundreds remain, which write in the hundreds' 2548 Minuend. 2325 Subtrahend. 223 Eemainder, SUBTRACTION. 41 place ; and 2 thousands from 2 thousands, nothing reuains. Consequently, 223 dollars is the answer required. I4 Proof of Subtraction. To prove the correctness of this, or any operation in sub- traction, add together the remainder and subtrahend. If this sum agree with the minuend, probably the operation is correct"; for the remainder and subtrahend, being the two parts into which the minuend is separated, the reunion of these parts ought to reproduce the minuend. ti^. Exercises in Subtracting when no Figure of the Subtrahend exceeds the Corresponding Figure of THE Minuend. In like Tnanner^ solve and explain the following problems. 1. Charles having 25 cents, gave 12 of them for a book. How many cents had he left? 2. Charles paid 25 cents for a book and slate, 13 cents was the price of the slate, what was the price of the book ? 3. John said he was 25 years younger than his father, who was 37 years old. How old was the boy ? 4. A merchant 35 years old, had traded 14 years. How old was he when he commenced business ? 5. In a school of 84 scholars, only 33 are girls. How many boys in that school ? 6. A man sold a chaise and harness for 198 dollars ; but the price of the chaise was 163 dollars. What was the price of the harness ? 7. A house and the land on which it stood cost 2350 dol- lars ; but the house cost all but 350 dollars. What was the cost of the house ? 8. If I deposite in a bank 1675 dollars, and afterv/ards draw out 1000, how much have I then remaining in the bank? 9. Mr. Walkers farm is worth 3000 dollars, and Mr. Dole's farm is worth 2000 dollars ; if they exchange farms, what should Mr. Dole pay Mr. Walker ? 10. What is the difference between 5643 and 643 ? 11. How much more is 12345 than 2040? 12. How much less is 1620 than 1840 ? -^ 13. Subtract 203040 from 516273. 1^ '* \ 42 ARITHMETIC. tl6* Model of a Recitation. 1. A man paid 85 dollars for a watch ; but was obliged to sell it for 67 dollars. What was his loss ? He lost the difference between what he gave, and what he received for his watch. Arrange the numbers and proceed as before directed. 85 7 units, however, cannot be taken from 5 units. 67 But, since (10) 1 unit of any order, equals 10 units of the next lower order, reduce (208) one of 18 the 8 tens to units, making 10 units, which, united with the 5 units, make 15 units, from which take the 7 units, 8 units remain, which write in the units' place, and take the 6 iens^not from 8 tens, for one of them has been reduced to units and disposed of; but take 6 tens from 7 tens, 1 ten remains, which write in the tens' place. Hence, 18 dollars is the answer required. 2. What is the difference between 9342 and 5739 ? Reduce one of the 4 tens to units, making ten units, which, united with the 2 units, make 12 units, 9342 from which take the 9 units, 3 units remain ; 5739 take the 3 tens of the subtrahend from the other 3 tens of the minuend, nothing remains ; there- 3603 fore, write a cipher in the tens' place ; reduce one of the 9 thousands to hundreds, making 10 hundreds, which, united with the 3 hundreds, make 13 hun- dreds, from which take the 7 hundreds, 6 hundreds remain ; take the 5 thousands from the other 8 thousands, 3 thou- sands remain. Hence, the whole difference is 3603. 57. Exercises in Subtracting, when some Figures .of THE Subtrahend exceed the Corresponding Figures OF the Minuend. In like manner, solve and explain the folUrwing problems. 1. A man gave 5 dollars for a hat, and 20 dollars for a coat. How much less did his hat cost than his coat ? 2. Dr. Franklin died A. D. 1790, and was 84 years old. In what year was he born ? 3. George Washington was born A. D. 1732, and died in 1799. How old was he when he died ? 4. The Puritans landed at Plymouth in 1620. How many years since ? SUBTRACTION, 43 5. How long since Columbus discovered America in 1492 ? 6. How many years since the declaration of Independence by the United States in 1776 ? 7. The Rocky Mountains are 12500, and the Andes 21440 feet high ; how much higher are the Andes than the Rocky Mountains ? 8. The Mississippi river is 3600 miles long, and the Missouri river is 4500 miles long ; how much longer is the latter than the former ? 9. In Massachusetts are 7800 square miles, and in New Hampshire 9491 ; how much more land in New Hampshire than in Massachusetts ? 10. How^much larger is New York, which contains 46085 square miles, than Massachusetts, which has 7800 square miles ? 11. Subtract 147 from 222. 12. From 671 take 584. 13. How much is 746475? 14. What must be added to 999, to make 1492 ? 15. What must be subtracted from 1840, to leave 1776 ? 58. Model of a Recitation. ^1. A man obtained at a bank, 300 dollars, but at the same time, he paid back 18 dollars for interest ; how many dollars had he left? He had left the difference between what he received and what he paid back, which is ascertained by subtracting 18 from 300. Here there are no units from which to take the 8 units, neither is there any ten to reduce to units ; there- ^f\r\ fore, reduce one of the 3 hundreds to tens, (SQ^) ^^ making 10 tens ; leaving 9 of these tens, reduce the other to units, making 10 units, from which ooQ take the 8 units; 2 units remain. Take the 1 ten in the subtrahend, from those 9 tens that you left unused ; 8 tens remain. There is nothing to take from the other 2 hundreds ; therefore, write them in the hundreds' place. Hence, 282 dollars is the answer required. 2. Subtract 30206, from 5000000. 44 ARITHMETIC. Reduce one of the 5 millions to hundred-thousands, making 10 ; one of which, (leaving 9,) reduce to ten- 5000000 thousands, making 10 ; one of which, (leaving 80206 9,) reduce to thousands, making 10 ; one of which, (leaving 9,) reduce to hundreds, making 4969794 10; one of which, (leaving 9,) reduce to tens, making 10 ; one of which, (leaving 9,) reduce to units, making 10 units, from which subtract the 6 units ; 4 units remain. Subtract the other figures of tbe subtrahend from the 9s that were left ; saying, cipher from 9 . tens leaves 9 tens ; 2 hundreds from 9 hundreds leaves 7 hun- dreds ; cipher from 9 thousands leaves 9 thousands ; 3 ten- thousands from 9 ten-thousands leaves 6 t(^-thousands ; blank from 9 hundred-thousands leaves 9 hundred-thousands ; and blank from 4 millions leaves 4 millions. Hence, the whole remainder is 4969794. 59. Observation. Observe, in these operations^ that the units of each order in the subtrahend^ beginning with the lowest^ are subtracted from the units of the same order ^ in the minuend^ when possible ; otherwise^ one of the units expressed by the next higher digit iii the minuend, is mentally reduced (leaving 95 in the intervening places) to the order of the deficient figure, and united with it, ivhen the subtraction is made from what then remains hi the several places of the min^ uend. 60. General Exercises in Subtraction. I?i like manner, solve and explain the following problems, 1. The top of a flag-staff, 25 feet long, which was fasten- ed to the top of a liberty-pole, was 104 feet high ; how high was the liberty-pole ? 2. If 17 feet should be broken from the top of a tree, 100 feet high, how high would be the stump ? 3. The bell on a church is 75 feet from the ground, but the vane is 102 feet from the ground ; how many feet from the bell to the vane ^ 4. If the Creation was 4004 years B. C, and the Deluge 234S years B. C, how man}^ years from the Creation to the Deluge ? 5. How many years from the Creation, 4004 years B. C. was Saul made the first king over Israel, in 1095, B. C. ? SUBTRACTION. 45 6. In 1820, New Orleans had 27176 inhabitants; in !.S25, 35000 inhabitants; what was the increase in five '^ears ? 7. In A. D. 1825, New Orleans had 35000 inhabitants ; in .830, 46310; what was the increase in five years ? 8. In A. D. 1830, New Orleans had 46310 inhabitants ; in 1835, 60000; what was the increase in these five years? 9. In A. D. 1835, New Orleans had 60000 inhabitants ; Jind Charleston, S. C., had 34500; how many more inhabi- tants in New Orleans, than in Charleston, S. C, in 1835? 10. In A.D. 1820, Philadelphia had 119325 inhabitants; in 1825, 140000 ; what was the increase in these five years ? 11. In A. D. 1825, Philadelphia had 140000 inhabitants; jn 1830, 167811 ; what was the increase in these ^ve years ? 12. In A.D. 1830, Philadelphia had 167811 inhabitants; :.n 1835, 200000 ; what was the increase in these five years ? 13. In A. D. 1820, Boston had 43298 inhabitants ; in 1825, 58277 ; what was the increase in these five years ? 14. In A.D. 1825, Boston had 58277 inhabitants; in :1830, 61381 ; what was the increase in these five years ? 15. In A.D. 1830, Boston had 61381 inhabitants; in 1835, 78613 ; what was the increase in these five years ? 16. In A. D. 1820, New York city had 123706 inhabitants ; in 1830, 203007 ; what was the increase in these ten years ? 17. In A.D. 1835, New York city had 269873 inhabi- tants; Boston had 78613; how many more inhabitants had New York than Boston ? 18. How much farther through the middle of the sun than through the middle of the earth ; the former being 883217 miles, and the latter being 7916 miles? 19. What is the diiTerence between the diameters of the Earth and Jupiter ; the former being 7916 miles, and the latter 89170 miles ? 20. How much faster does the Earth move than Jupiter ; the former moving 68000 miles an hour, the latter 30000 miles an hour ? 21. How much is 1000 999 ? 22. How much more is 380064 than 87065 ? 23. How much smaller is 8756 than 37005078? 24. How much must you add to 7643, to make 16487 ? 25. How much must you subtract from 2483, to leave 527 ? 26. What is the difference between 487068 and 24703 ? k 46 ARITHMETIC. 27. If you divide 3880 dollars between two men, giving one 1907 dollars ; how much will you give the other ? 28. Subtract 2222 from 3111. 29. Subtract 9 from 1000. 30. Seven millions, minus seventeen, is how much ? V. DIVISION. 61 The Principles of Division Illustrated. 1. A butcher having 35 sheep, began Monday morning, and killed 5 every morning as long as they lasted ; how many days did they last ? Since he killed 5 sheep each day, they would last as many days as there are times 5 sheep in 35 sheep. After he had killed 5, Monday, 30 remained ; Tuesday, 25 remained ; Wednesday, 20 remained ; 35 Thursday, 15 remained ; Friday, 10 re- 5 Monday. mained; Saturday, 5 remained; and, Sunday, he killed the last 5 ; and none 30 remained. Hence they lasted 7 days. 5 Tuesday. But when it is to be ascertained how many times a given number can be 25 subtracted from another given number, 5 Wednesday, that is, how many times a subtrahend is contained in a minuend, it can be done 20 by a shorter process than subtracting once 5 Thursday, the subtrahend at a time. Write 35, the minu- 15 5) 35 (7 days. end ; draw a line on 5 Friday. 35 each side, to distin- guish it from the other 10 00 numbers to be written 5 Saturday. with it, and at the left hand, write 5, the subtrahend. Now, 5 think how many 5s there are in 35, and 5 Sunday. place the number at the right hand. To ascertain whether you thought the right number, subtract so many times 5 all at once. If there is nothing left, your num- ber is right ; for, if there are exactly 7 fives in 35, then the sum. of 7 times 5, subtracted from 35, should leave nothing. DIVISION. 47 "teacher having 48 scholars studying arithmetic, separated them into classes of 12 scholars each; how many dosses did he nt&ke ? Since he put 12 scholars into each class, he would make as many classes as there are times 12 scholars in 48 scholars. Write the 48 ; draw a line on each 12) 48 (4 classes, side; and write the 12 at the left hand. 48 Now, how many 12s do you think there are in 48 ? Four 12s. Very well ! 00 Place the 4 at the right hand, and ascer- tain whether 4 such classes take exactly all of the 48 scholars. 3. A butcher killed 35 sheep in 7 days ; how many would ihat be each day ? Killing one each day would require 7 sheep ; therefore, he would kill as many each day, as he had times 7 sheep. Write the 35, draw the lines, and write the 7 at the left hand, think how many 7s there 7) 35 (5 sheep a day. are in 35, and place the number 35 at the right hand. This num- ber is the answer required, if 7 00 multiplied by it make exactly 35. 4. A teacher having 48 scholars studying arithmetic, separated them into 4 equal classes ; how many could he put into each class ? Putting one into each class would require 4 scholars ; therefore, he could put as many into each diss, as he had times 4 scholars. Arrange the tvo given 4)48(12 scholars a class. numbers, think how many 48 4s there are in 48, ana place the number at the right hand 00 for the answer required. Then ascertain whether 4 multiplied by this number, take exactly all the scholars. r. 66. 1. How many quarts are there in 600 pints? Since there are 2 pints in a quart, there will be as many quarts as there are times 2 pints in 600 pints. DIVISION. , 61 2 is contained 3 times in 6 units of the first order, but, in 6 units of the third order, which 2 ) 600 ( 300 quarts. are 100 times as large, (6,) it 600 must be contained 100 times as often, which is 300 times. 300 quarts, at 2 pints each, take 600 )ints, which subtracted from 600 pints, nothing remains, rlence, 300 quarts is the answer required. ^Sy. Exercises in Dividing Units of any one Order. In like manner, solve and explain the following prob- lems, 1. If in a certain school-room 2 scholars sit at a desk, !iow many desks will accommodate 200 scholars ? 2. If in a certain school there are 80 scholars, and 2 ;eachers, how many scholars are there for each teacher ? 3. If a man pay 3 dollars apiece for hats, how many hats can he buy for 90 dollars ? 4. If Mr. Farmer sell 2 cows for 40 dollars, how much is i:hat apiece ? 5. At 4 dollars a yard for cloth, how many yards can be bought for 80 dollars ? 6. If 800 dollars a year be paid to 4 female teachers, hoi?^ much is that apiece ? 7. At the rate of 5 miles an hour, how long would it take to travel 500 miles ? 8. If 6 shares in a bank cost 600 dollars, how much is that a share ? 9. At an average of 7 persons to a family, how many families in a town of 7000 persons ? 10. If 90000 dollars be the cost of 3 miles of rail-road, what is the cost per mile ? 68. Model of a Recitation. 1. A hatter made in a year 560 hats, and packed them foi market in boxes holding 8 hats apiece. How many boxes would he need ? Since each box would hold 8 hats, he would need as many boxes as there are 'times 8 hats in 560 hats. But, one unit GZ ARITHMETIC. of any order making ten units of the next lower order, (10) the 5 hundreds are equal to 50 tens, 8 ) 560 ( 70 boxes. which with the 6 tens, make 56 560 tens ; 8 is contained 7 times in 56 units of the first order, but in 56 units of the second order, which are 10 times as large, (Aj) it must be contained 10 times as often, which is 70 times ; 70 boxes at 8 hats each, would take 560 hats, which subtracted from 560 hats, nothing remains. Hence, 70 boxes is the answer required. 69* Exercises in Reducing Units of a High to a Lower Order for Division. In like manner^ solve and explain the following prob' lems, 1. How many pairs of boots could be bought for 150 dol- lars at 3 dollars a pair ? 2. If 350 dollars be paid for 5 horses, how much is that apiece ? 3. How many hours will it take to travel 350 miles at 7 miles per hour ? 4. If a stage travel 120 miles in 12 hours, how far is that an hour ? 5. How many times is 5 contained in 450 ? 6. Into how many parts of 9 each can 6300 be divided ? 7. If 3200 be divided into 8 equal parts, how large are the parts ? 8. Divide 2500 into 5 parts ; how large is each part ? 9. What number must 7 be multiplied by to produce 4900 ? 10. What number multiplied by 3 will produce 27000 ? 11. Divide 100 by 4. 12. Divide 1000 by 8. 13. If ISOO be the dividend and 9 the divisor, what will be the quotient ? TO. Explanation of the Written Process of Division. 1. How many yards are there in 9636 feet ? Since there are 3 feet in a yard, there will be as many yards as there are times 3 feet in 9636 feet. ' DIVISION. S3 3 ) 9636 ( 3000 yards. 200 yards. 10 yards. 2 yards. 3212 yards. 9000 636 600 36 30 6 6 3 is contained 3 times in 9 units of the first order, but in 9 units of the fourth order, it must be contained 1000 times as often, (lO,) that is, 3000 times; 3000 yards at 3 feet each, take 9000 feet, which sub- tracted from 9636 feet, leave 636 feet; 3 is contained in 6 2mits 2 times ; therefore, in 6 hundreds, it is contained 200 times ; 200 yards at 3 feet each take 600 feet, which subtracted from 636 feet leave 36 feet ; 3 is contained in 3 U7iiis 1 time, therefore, in 3 tens it is con- tained 10 times; 10 yards at 3 feet each take 30 feet, which subtracted from 36 feet leave 6 feet, in which 3 is contained 2 times ; 2 yards at 3 feet each take 6 feet, which subtracted from 6 feet, nothing remains. Hence, 3000 yards -|- 200 }ards -j- 10 yards -}- 2 yards = 3212 yards, is the answer required. This operation may be abridged by omitting some unnec- essary figures. Instead of the ciphers belonging to the first number in the quotient, write the digits of the other numbers as they are obtained, which will finally leave each figure in its own place. The product of the divisor and the first quotient figure is 9 thousand ; omitting the ciphers, it will be sufficient to write the 9 in the thousands' place, and subtract it from the thousands ; then bring down the- 6 hundreds only, for consideration ; 200 times the divisor is 6 hundreds, which being subtracted from the hundreds, bring down the 3 tens ; 10 times the divisor is 3 tens, wMch being subtracted from the tens, bring down the 6 units ; 2 times the divisor is 6 units, which being subtracted from the units, nothing more 5* 3 ) 9636 ( 3212 yards. 9-- 6- 6- 3- 3- 6 6 54 ARITHMETIC. of the dividend remains. Hence, 3212 yards is the answer required, as before. 71, Model of a Recitation. Divide 2848 by 4, or find how many times 4 is contained in 2848. 4 is contained 7 times in 28 units, but in 28 hundreds it is contained 100 times as often, (68^) or 4 )2848{ 712 7 hundred times; 7 hundred times 4 28 are 28 hundred, which subtract from the hundreds, and bring down the 4 4 tens ; 4 is contained 1 time in 4 units, but 4 in 4 tens it is contained 10 times as often, or 1 ten times ; 10 times 4 are 8 4 tens, which subtract from the tens and 8 bring down the 8 units ; 4 is contained 2 times in 8 units ; 2 times 4 are 8, which subtracted, nothing remains ; conse- quently, 712 is the result required. 72. Exercises in Explaining the Written Process of Division. In like manner, solve and explain the follounng probhrns, 1. How many bushels in 88 pecks ? 2. How many weeks in 77 days ? 3. How many dollars in 126 shillings ? 4. If 4 horses are required to draw 1 wagon, how many wagons might be drawn by 168 horses ? 5. If a man can travel 5 miles an hour, how many hours would it take him to travel 205 miles ? 6. A drover received 248 dollars for sheep that he sold for 4 dollars a head. How many were there ? 7. If 5 bushels of corn pay for a pair of boots, how many pairs would 255 bushels pay for ? 8. Suppose 6 men should contribute 186 dollars, how much would that be apiece ? 9. Suppose 355 dollars' bounty were paid at 5 dollars apiece to a company of soldiers. How many soldiers in the company. 10. How many weeks can a man get board for 156 del lars, at 3 dollars a week ? 11. How many times is 7 contained in 637? mm-. DIVISION^ 66 Stippose 3699 to be a dividend, and 9 a divisor, what is the quotient? 13. Divide 1S36 by 3. 14. What must I multiply by 8 to make 7288 ? 15. Into how many parts of 5 each can 555 be divided ? 16. If 567 be divided into 7 equal parts, what must be the size of each part? 73. Model of a Recitation. Mr. Farmer planted 4785 grains of corn in a field, planting I'i grains in each hill. How many hills did he make ? Since he put 5 grains in each hill, he made as many hills IS there are times 5 grains in 4785 grains. Beginning at the left hand of the dividend, take into con- sideration the fewest figures that 5 ) 4785 ( 957 hills. can contain the divisor ; as 5 is not 45 contained in 4, take 47 hundreds, in 45 of which 5 is contained 9 28 hundreds times, (lO,) 900 hills 25 require 45 hundred grains, which subtracted from 47 hundred leave 35 2 hundred, with which join the 8 35 ^ tens, making 28 tens, (IO5) in 25 of which 5 is contained 5 tens times ; 50 hills require 25 tens grains, which subtracted from 28 tens leave 3 tens, with which join the 5 units, making 35 units, in which 5 is contained 7 times ; 7 hills require 35 grains, which subtracted from 35 grains, nothing remains. Hence, 957 hills is the answer required. 74. Observation. Observe, (73^) that the division is comm?iced, by dividing the fewest figures on the left of the dividend that will contain the divisor, that the quotient figure will be of the same denomination as that part of the dividend from which it is obtained^ that each succeeding figure of the dividend will require an additional figure in the quotient, a cipher if nothing larger, that the products of the divisor, by each quotient figure, are to be subtracted from those parts of the dividend from lohich the respective quotient figures are obtained, that the remainder in each case is reduced (208) and united to the units of the next lower order ^ for division^ 56 ARITHMETIC. and that the sum of these partial products^ or the product of the divisor by the whole quotient^ is equal to the dividend. 75. Proof of Division. To prove the correctness of an operation in Division, multi- ply the divisor and quotient together ; if their product equals the dividend, probably the operation is correct ; for, the cor- rect quotiem, expressing how many times the divisor there are in the dividend, (61,) is one, and the divisor the other of two factors, whose product should be the dividend. 76. Exercises requiring some Units of each Order to BE reduced to a Lower Order for Division. In like manner, solve and explain the following problems. 1. If 9 hills of potatoes yield a bushel, how many bushels of potatoes in a field of 1296 hills ? 2. If an army of 2048 men were marching in sections, having 8 men in each section, how many sections would be there ? 3. If in an army every ninth man is an officer, how many officers in an army of 4608 men ? 4. If a general should divide his army of 12096 men into 7 equal divisions, how many men would be in each division ? 5. How many weeks in 364 days ? 6. How many Sabbath days in 12852 days ? 7. If an acre of land pasture 5 sheep, how many acres could pasture 315 sheep ? 8. How many times is 6 contained in 738 ? 9. How many times is 4 contained in 20012 ? 10. Divide 3606 by 3. 11. Divide 25634 by 2. 12. If 28028 be a dividend, and 7 a divisor, what is the quotient ? 13. If 18675 be a product, and 5 one factor, what is the other factor ? 14. What must 11889 be divided by, to give 9 for a quo- tient ? 15. "What must 8 be multiplied by, to produce 2496 ? 77. Model of a Recitation. 1. If in the month of July a rail-road company received DIVISION. 57 6284 dollars from passengers, at 2 dollars apiece, how many passengers rode in the cars in that month ? Since each passenger paid 2 dollars, there were as many passengers as there are times 2 dollars in 6284 dollars. To obtain the answer by a still shorter process, write the cividend and divisor as heretofore, but perform the operation ii your mind, writing only the quotient, and write that under the dividend, with each figure under that of its own order. Thus, 2 in 6 thousands 3 thousand 2) 6284 times, therefore, write 3 in the thou- sands' place ; 2 in 2 hundreds 1 3142 passengers. hundred times, therefore, write 1 in the hundreds' place ; 2 in 8 tens 4 tsns times, therefore, write 4 in the tens' place ; 2 in 4 units fil times, therefore, write 2 in the units' place : making 3142 times 2 dollars. Hence, 3142 passengers is the answer re- quired. 2. If a stage run 6 miles an hour, how many hours would i^. take the stage to run 1848 miles ? Since in one hour it runs 6 miles, it will take as many hours *as there are times 6 miles in 1848 miles. 6 in 18 hundreds 3 hundreds times ; write 6 ) 1848 3 in the hundreds' place. If 6 were con- tained in the 4, which is tens, the quo- 308 hours. tient figure would be tens, but as 6 is not contained in 4, there are no tens in the quotient, therefore, write a cipher in the tens' place, and reduce the 4 tens to units, making 40 units, which, joined with the 8 units, make 48 units, in which 6 is contained 8 times, therefore, write 8 in the units' place : making 308 times 6. Hence, 308 hours is the answer required. 78. Exercises in Abridging the Process of Division. In like jnanner, solve and explain the following problems. 1. If 306 dollars be divided among 3 men, what is each man's share ? 2. If 4 shares of a bank cost 416 dollars, what would one share cost ? 3. If six brothers receive a legacy of 1512 dollars, what would be the share of each ? 4. Paid 150 dollars for 6 tons of hay. How much was that for a ton ? I 53 ARITHMETIC. 5. If there are 1280 inhabitants in a town, and the families average 8 persons apiece, how many families in that town ? 6. How many yards of cloth can be bought for 1155 dol- lars, at 7 dollars a yard ? 7. Find a nmnber, which, multiplied by 9, will produce 63234. 8. What number, multiplied by 8, will produce 2464 ? 9. What number, divided by 9, will give 72 for a quotient ? 10. If 7 be a divisor, and 42014 a dividend, what is the quotient ? 11. How many times is 5 contained in 1204500890 ? 12. How many times does 540010 contain 5 ? 13. How many times 8 are there in 25648 ? 14. Divide 4004 by 4. 15. Divide 16800 by 8. 16. Divide 36900 by 3. 'WtH^ 17. Divide 1800108 by 9. WW 18. Divide 105105 by 7. 19. If 1836 be a dividend, and 9 the divisor, what is the quotient ? 20. If 1728 be divided by 9, what would be the quotient ? 21. If 72 be a dividend, and 9 the quotient, what is the divisor ? 22. If 63 be a dividend, and 7 the quotient, what is the divisor ? 79 Model of a Recitation. 1. How many days in 1728 hours ? Since in one day there are 24 hours, there must be as many days as there are times 24 hours in 1728 hours. 24 is contained in 172 tens 7 tens 24) 1728 (72 days. times ; 70 times 24 make 168 tens, 168 which, subtracted from 172 tens, leave 4 tens, to which bring down the 8 48 units, making 48 units, in which 24 48 is contained 2 times ; 2 times 24 make 48, which subtracted from 48, nothing remains. Hence, as there are 72 times 24 hoiirs, 72 days is the answer required. 2. How many times is 64237 contained in 436940074 ? The many figures in thfs divisor, present a difficulty in ascertaining any quotient figure. The best way is to seek Divisioif. 59 hjw many times the highest figure only, of the divisor, is cmtained in the highest one, or two, figures of the dividend ; this quotient figure will either be right, or one or two too krge ; for the greater certainty, however, before multiplying the whole divisor by it, multiply mentally only one or two of the highest figures of the divisor, and compare the product v^ith the highest figures of the dividend from which this part o the product is to be subtracted ; if the appearance is satis- fy -ctory, proceed with this quotient figure, otherwise take a smaller figure, and proceed. If at any time a product prove too large to be subtracted, the last quotient figure is too large ; or, if a remainder be k.rger than the divisor, the last quotient figure is too small. Ill either case, erase it, and try another figure. 6 is contained 7 times in 64237) 436940074 (6802 times. 43, but 7 times 64 is 385422 greater than 436; there- fore, 7 is too large for the 515180 first quotient figure ; write 513896 6 in the quotient, and subtract 6 thousand times 128474 the divisor, that is, 6 times 128474 the divisor from the thou- sands, and to the remain- der bring down the next figure of the dividend ; 6 is contained 8 times in 51, and 8 times 64 being less than 515, subtract 8 hundred times the divisor," that is, 8 times the divisor from these hundreds, and to the remainder bring down the next figure ; this number being smaller than the divisor, there can be no tens in the quotient; therefore, write a cipher in the tens' place, (77'^) and bring down the next figure ; 6 is contained in, 12 twice ; subtract 2 times the divisor, and nothing remains. Hence, 6802 times, is the answer required. 80, General Exercises in Division. In like manner, solve and 'explain the following problems, 1. How many days in 360 hours ? 2. If a man travel 45 miles a day, in how many days will he travel 1125 miles ? 3. A butcher gave 875 dollars for 35 cows. What was the t of each cow ? f M ARITHMETIC. 4. If a field of 34 acres produce 1020 bushels of corn, how much would that be per acre ? 5. Suppose an acre of land to produce 38 bushels of corn, how many acres must be cultivated to produce 4902 bushels ? 6. How many horses, at 75 dollars apiece, can be bought for 1125 dollars? 7. A school-district paid a teacher 144 dollars for teaching, at 36 dollars a month. How long was the school kept ? 8. If a man's income be 1095 dollars for 365 days, how much is that per day ? 9. How many hogsheads, of 63 gallons each, can be filled from 8379 gallons ? 10. How many years in 8395 days, if 365 days be called a year ? 11. If 1512 dollars be divided among some brothers, so that each may receive 252 dollars, how many are the brothers ? 12. How many bank shares can be purchased with 2912 dollars, at 112 dollars each ? 13. How many acres of land will yield 6996 bushels of potatoes, if 212 bushels grow on one acre ? 14. How many barrels must a man have to fill from 125440 pounds of flour, if each barrel hold 196 pounds ? 15. A man put 17484 pounds of tea into 186 chests. How much in each chest ? 16. How many times can 48 be subtracted from 5040 ? 17. How many times is 75 contained in 23025 ? 18. How many times 25 is equal to 23025 ? 19. How many times does 105735 contain 105 ? 20. How many times does 105735 contain 1007 ? 21. Divide 144144 into 144 equal parts ; what is each part ? 22. Divide 172800 nuts among some boys, giving them 1440 nuts apiece. How many boys can you supply with them ? 23. What number, multiplied by 754, will produce 18850 ? 24. The product of two factors is 612060. If one factor is 303, what is the other factor ? 25. Divide a city of 78612 inhabitants into 12 equal wards. How many inhabitants in each ward ? 26. How many equal parts can be made of 1048576, if 1024 be one of the parts ? 27. How many times 409^ is equal to 262144 ? DIVISION. 61 28. If 2048 be one of a certain number of equal parts of ^ 31 072, how many are the parts ? \it. General Exercises in the Fundamental Principles OF Arithmetic. 1. There are two numbers, of which the greater is 27 times the less, and the less is contained 9 times in 27. What are the numbers ? 2. A was born when B was 26 years old. How old will A be when B is 45 ? 3. If the sum of 3 numbers be 500, the difference between tlie least and the greatest be 174, and the difference between tlie middle number and the sum of the 3 numbers be 350, what are the numbers ? 4. A man bought 5 pieces of cloth at 44 dollars each, 974 piirs of shoes at 2 dollars a pair, 600 pieces of calico at 6 dDllars each, and sold the whole for 6000 dollars. How niuch did he gain, or lose ? 5. A man exchanged 6 cows at 15 dollars each, a yoke of oxen at 67 dollars, for a horse at 50 dollars, and a chaise. What did the chaise cost ? 6. A boy bought some apples, and, after giving away 10, and buying 34 more, he divided half of what he then had among 4 companions, giving them 8 apiece. How many apples did he buy at first ? 7. What is that number, to which, if 4 be added, from which 7 be subtracted, the remainder multiplied by 8, and the product divided by 3, the quotient will be 64 ? 8. A man bought a farm at 25 dollars an acre, and sold half of it, at the same rate, for 1850 dollars. How many acres did he buy ? 9. Five men and three boys were paid a sum of money, so large that each man had 43 dollars, and each boy 25 dollars. What was the whole sum ? 10. If a trader gain 160 dollars on 544 barrels of flour, that cost him 6 dollars a barrel, besides 25 dollars that he paid for storage ; what would he receive for the flour? 11. Suppose 5 bushels of wheat make a barrel of flour, how many barrels can be made from the wheat raised on 75 acres, at 29 bushels per acre ? 12. How many times 6 in 75 tjmes 29 ? 13. A farmer exchanges 44 acres of land, worth 36 dollars 6 I 62 ARITHMETIC. an aero, for 66 acres of land in another place. What does his land cost him per acre ? 14. A man who owned 520 acres, bought 375 acres more, and, reserving 95 acres for himself, divided the remainder into 8 equal farms, and sold them for 2500 dollars apiece. How much did he get per acre for his land ? 15. If a man's income be 1349 dollars a year, and his ex- penses 20 dollars a week, how much would he save in a year? 16. A merchant's business brought him, in a year, 2500 dollars ; but his expenses were 1772 dollars. How much did he save per week ? 17. If I buy 245 hogsheads of molasses, at 18 dollars each, how much do I gain, or lose, in selling it for 4000 dollars ? 18. If a man's expenses be 2 dollars a day, and his income 17 dollars a week, how many weeks will it take him to save 156 dollars ? 19. If a lot of land be divided into 8 farms, each of 150 acres, and the farms be sold for 3000 dollars apiece, what would one acre cost ? 20. A gentleman bought 2 pieces of land, one contained 96 acres, the other 103 acres. If he should sell 47 acres, at 25 dollars an acre, how much would the rest of the land be worth at the same rate ? 21. A merchant bought a cask of molasses containing 119 gallons, and sold to one man 10 gallons, to another 9 gallons, to another 25 gallons. How much is the remainder worth, at 40 cents a gallon ? 22. What is the difference between 17 times 105 and 3417 divided by 17 ? 23. What is the difference between 20 times 210 and 7 times 2500 divided by 175 ? 24. If I purchase 1200 pounds of butter for 15600 cents, how must I sell it per pound to gain 2400 cents ? 25. If I buy 375 pounds of pork at 7 cents a pound, and sell it for 3000 cents, how much do I gain on a pound ? 26. How many quintals of fish, at 2 dollars each, will pay for 500 hogsheads of salt, at 5 dollars a hogshead ? 27. How much flour, at 7 dollars a barrel, will pay for 224 cords of wood, at 8 dollars a cord ? 28. How many days must 3 brothers work to receive 2475 DIVISION. 63 celits, if one earn 42 cents a day, the second 32 cents, and ;he youngest 25 cents ? 29. If a man earn 6 dollars a week, and his two boys earn 3 dollars apiece a week, how many weeks will it take them ill to earn 624 dollars ? 30. If a hogshead hold 252 quarts, and two boys work :ogether to fill it with water, one having a pail which holds 12 quarts, the other having a pail which holds 9 quarts, how nany times must they empty their pails to fill the hogshead ? 31. If a full hogshead should begin to leak in 3 places, at 3nce, from one hole 4 quarts a day, from another 2 quarts a day, and from the other 1 quart a day, how many days before the hogshead would be emptied ? 32. A man bought some sheep and calves, and of each an equal number, for 165 dollars, giving for the sheep 7 dollars apiece, and for the calves 4 dollars apiece. How many were there of each sort ? 33. How many coats, pantaloons and vests, of each an equal number, can be made from 405 yards, if it take 5 yards for a coat, 3 yards for a pair of pantaloons, and 1 yard for a vest ? 34. If 9000 men march in a column of 750 deep, how many inarch abreast ? 35. A man left his estate, valued at 8956 dollars, to his wife and daughters, giving his wife 4688 dollars, and his daughters 1067 dollars apiece. How many daughters had he ? 36. The factors of a certain number are the diiTerence be- tween 1632 and 1700, and between 94 and 5 dozen. What is that number ? 37. Paid 57600 cents ^r eggs, at 12 cents a dozen. How many eggs did I buy ? 38. A boy bought a sled for 96 cents, exchanged it for 8 quarts of nuts, sold half of his nuts at 12 cents a quart, and gave the rest of his nuts for a penknife, which he sold for 34 cents. How much did he gain, or lose ? 39. Three men owned farms situated together ; the first had 64 acres, the second had 20 acres more than the first, and the third had as many acres as both the first and second ; the three farms were worth 7400 dollars. What is that per acre ? 40. If a man owe 728 dollars to Mr. Saveall, and works for him to pay the debt ; how many years, of 52 weeks each, will it take him, if he pay only one dollar a week ? 64 ARITHMETIC. 41. If a man earn 40 dollars a month, and spend 13 dol- lars of it each month, how long will it take him to pay for a house worth 1620 dollars ? 42. A farmer sold some pork at 17 dollars a barrel to the amount of 510 dollars, and some at 19 dollars a barrel to the amount of 380 dollars, how many barrels did he sell ? 43. A drover exchanged 42 horses worth 72 dollars apiece, for cows worth 36 dollars apiece, and for his cows he received 36 yoke of oxen, which he sold so as to gain 144 dollars, how much did he get for each yoke of oxen ? 44. How much is 72 X 24 36 + 84 X 7 -5- 12 11 ? 45. Let 27 be a divisor, and 567 a dividend, what will be the quotient ? 46. Suppose 25 is a quotient, and 25 a divisor of the same dividend, what is that dividend ? 47. Of what dividend is 15 both divisor and quotient ? VI. FRACTIONS. 82. Origin of Fractions, and Manner of Writing them. 1. At 5 cents a quart for nuts, how many quarts can you buy for 38 c^nts ? Since 1 quart costs 5 cents, you can buy as many quarts as there are times 5 cents in 38 cents. 5 is contained 7 times in 35, which being subtracted from 38, there remain 3 units, which are 5 ) 38 ( 7-| quarts. not sufficient to contain the whole of 35 5, but if %f be divided into 5 equal parts, each part is exactly a unit ; 3 therefore, your remainder being 3 3 units, will contain exactly 3 of the parts of 5, which being subtracted, nothing remains. In the quotient write 7 to express the number of whole 5s ; after it near the top write a small 3 to express the number ofparts^ and under the 3, separated by a line, write a small 5, to express the size of these parts, which it will do by showing how many such parts make a zmit, or a whole, of which these are parts. Hence, as 38 contains 7 times 5 and 3 such parts of a 5 that 6 of these parts would make a whole 5, you can buy 7 whole FRACTIONS. 65 cuarts and 3 such parts of a quart that 5 of them would riake a whole quart. Another Explanation. 35 cents will buy 7 quarts, and you have 3 cents remain- ing, which are not sufficient to buy a 5)38 whole quart: but if a whole quart be divided into 5 equal parts, each part will I 7f quarts. be worth exactly one cent ; and as you I have 3 cents remaining, you can buy 3 of f these parts. Hence, with your 38 cents, jou can buy 7 whole quarts snd 3 such parts of a quart that 5 of them would make a %irhole quart. After the 7 (whole quarts) near the top, write a small 3 to express the number of parts, and under the 3 separated by a line, write a small 5 to express the size of these partSy "^v^hich it will do by showing how many such parts make a unit, or whole quart. 2. If 25 apples be given to 7 boys, what would be the share of each boy ? Since giving one apple to each boy takes 7 apples, the share of each boy would be as many apples as there are times 7 apples in 25 apples. 7 is contained 3 times in 21, which being subtracted from 25, there remain 4 units, which are 7 ) 25 ( 3f- apples. not sufficient to contain the ivhole of 21 7 ; but if 7 be divided into 7 equal I; parts, each part will be exactly a W 4 unit; therefore the remainder, being F 4 4 units, will contain exactly 4 of these parts of 7, which being sub- tracted, nothing remains. In the quotient write 3 to express the number of ivhole 7s ; after it near the top write a small 4 to express the number of parts, and under the 4, separated by a line, write a small 7 to express the size of these parts, which it will do by showing how many such parts make a unit, or a whole 7. Hence, as 25 contains 3 times 7 and 4 such parts of a 7 that 7 of them would make a whole 7, liie share of each boy would be 3 whole apples and 4 such parts of an apple that 7 of them would make a whole apple. 6=^ I S6 ARITHMETIC. Another explanation, 21 apples will afford 3 to each boy ; but the 4 remaining apples will not afford the boys a whole 7 ) 25 apple apiece. If, however, each of these 4 apples be cut into 7 equal parts, they 3f apples. would make 28 parts, or exactly 4 parts for each boy. Hence, the share of each boy would be 3 whole apples, and 4 such parts of an apple that 7 of them would make a whole apple. This answer may be expressed as before directed. 83. Definition of Terms. A fraction is the expression of one, or more of the eqtuil parts of a unit, A fraction is composed of two numbers, called the terms of the fraction. The terms of a fraction are written one below the other, separated by a line. The upper term called the numerator shows how many parts the fraction expresses. The lower term called the denominator shows the size of the parts expressed by the fraction, by showing how many such parts make the unit of which the fraction ex- presses one or more parts. 84. Manner of Reading Fractions and Mixed Numbers. Parts take different names ^ according to their size^ or the number of them that it takes to make a unit. Thus, the fractions f , f , ^^, y^, &c., are read, two thirds^ 2 fifths, two tenths, 2 hundredths, &c. A fraction may be considered and read in four different ways ; for instance, |- may be considered f of 1, or ^ of 3, or 3 divided by 4, or 3 such parts that 4 like them would make a unit. A number which is composed of both an integral and fractional number, is called a mixed number. The answers to the above problems (82) 7f , 3^, are mixed numbers, which are read thus, seven and three fifths, three and four sevenths. Integer is a term applied to a number which expresses only whole units. FRACTIONS. 07 ml^iSm Exercises in Originating and Writing Fractions. 11 Solve and explain the following prohleTus on the left^ like the firsts and those on the rights like the second of the above vxamples (82.) I. If 1 lead pencil cost 3 oents, how many can you buy or 8 cents ? 3. If 4 cents buy an orange, .low many can be bought for \IS cents ? 5. If the stage fare be 6 cents a mile, how far can you ride for 41 cents ? 7. How many slates at 8 cents apiece, can be bought for 93 cents ? 9. How many writing books at 10 cents apiece can be bought for 125 cents ? II. How many shad at 15 cents apiece, can be bought for 218 cents ? 13. At 22 cents for an inkstand, how many may be bought for 93 cents ? 15. At 29 dollars a head, how many cows maybe bought for 350 dollars ? 17. How many acres of land at 37 dollars an acre, will 5565 dollars buy ? 19. If 320 rods make a mile, how many miles in 46100 rods ? 21. If a ship sail 125 miles per day, ^how long would it take her to sail round the world, it being 24911 miles ? 2. If you divide 11 lead pencils among 3 boys, how many will each boy have ? 4. How many cents does 1 lemon cost, when you give 22 cents for 5 lemons ? 6. How much does a man earn a week, who receives 65 dollars for 7 weeks ? 8. If 9 men do a job together, and receive 220 dol- lars, what is the share of each? 10. What does a single knife cost, at 295 cents a dozen ? 12. What is the price of a barrel of flour, when 18 barrels cost 150 dollars ? 14. If 25 apple trees yield 183 bushels of apples, how much does each tree yield ? 16. If in 32 equal loads of potatoes 729 bushels were car^ ried to market, how many bushels in each load ? 15. If 1760 yards make 320 rods, how many yards make Irod? 20. If 1749 feet make 106 rods, how many feet in one rod? 22. If a ship sail 132 miles a day, in how many days will she sail Jrom Boston to Liver- pool, it being 3000 miles ? I ARITHMETIC. 24. What would be the cost of 1 hogshead of molasses, if 75 hogsheads cost 2200 dol- lars? 26. How many bushels of wheat does a farmer raise on an acre, who raises 2400 bushels on 99 acres ? 28. If a man receive 730 dollars a year, how much is that a week ? 30. How many miles per hour does an engine move, which goes 2600 miles in a week? 32. Divide 4657 into 25 equal parts. 34. Divide 100000000 by 12478. 23. If 63 gallons of water in one hour run into a cistern containing 432 gallons, in what time will it be filled ? 25. How many boxes would be required to contain 32844 oranges, if each box contain exactly 100 oranges ? 27. How many days, at 175 cents a day, must a man work to earn 4500 cents ? 29. At 365 days a year, how many years old is a boy who has lived 3999 days ? 31. How many times 199 in 2569? 33. Divide 2864 by 14. 8. Observation. Observe, (S^j) that when a number is to be divided which is smaller than the divisor, the quotient will be a fraction, of which the dividend will he the numerator and the divisor will be the denominator. Hence, division may be expressed in a fractional form, whether the dividend be larger or smaller than the divisor, and the value of the expression will be the true quotient. 87. Model of a Recitation. 1. If a pie be cut into 8 equal parts, what fractions would express one, three, five, and eight of the parts ? When 8 equal parts make a unit, any number of these parts are so many eighths ; (84) therefore, one part is -J- (one eighth,) three parts are f (three eighths,) 5 parts are |- (five eighths,) and eight parts are f (eight eighths,) or the whole pie. 2. What fractions of a foot will express 5, 7, and 11 inches ? When a unit is divided into 12 equal parts, any number of the parts are so many twelfths, (84,) therefore, 5 inches FRACTIONS. . 69 aie 1^ of a foot, 7 inches are xV of a foot, and 11 inches ai e \^ of a foot, 3. What parts of 15 are 8, 14, and 19 ? Since it takes 15 units to make the whole of 15, any num- b(;r of units are so many fifteenths of 15 ; therefore, 8 is -^ 01' 15, 14 is if of 15, and 19 is |f of 15. 88. Exercises in Expressing Division. In like manner, solve and explain the following problems, 1. If a pie be cut into 6 equal pieces, what fractions will express one, two, and five of the pieces ? 2. Two boys divided an orange equally between them- selves, what fraction will express each one's part ? * 3. If an acre of land be divided into 4 equal house-lots, what fractions would express one, three, and four of the lots ? 4. If a piece of cloth be sufficient for 7 coats, what parts 01 the piece of cloth would be sufficient for 1, 3, 5, and 6 coats ? 5. If you divide a barrel of flour equally among 9 men, what part of a barrel would each receive ? 6. If IS dollars be paid for a ton of hay, what parts of a ton may be bought for 5, 7, 11, and 17 dollars ? 7. At 27 dollars a hogshead for molasses, what parts of a hogshead may be bought for 10, 14, 19, and 25 dollars? 8. At one hundred dollars a share in a bank, what parts of a share may be bought for 16, 29, 67, 89, and 93 dollars. 9. At 75 cents a bushel for corn, what parts of a bushel may be bought for 12, 24, 36, and 58 cents ? 10. What fractions of a dollar will express 7, 23, 37, 47, 67, and 97 cents ? 11. What fractions of June will express 11, 17, and 29 days ? 12. What parts of July are 16, 21, and 27 days ? 13. What parts of an hour are 13, 43, and 59 minutes ? 14. What parts of a day are 1, 7, 19, and 23 hours ? 15. 11, 21, 87, 123, and 219 rods are what parts of a mile ? 16. What part of 5 dollars are 3 dollars ? 17. What parts of 25 cents are 3, 7, 14, and 21 cents ? 18. What parts of 63 gallons are 16, 31, and 44 gal- lons ? 70 ARITHMETIC. 19. What parts of 365 days are 31, 60, 124, 243, and 316 days ? 20. 15 weeks are what part of 52 weeks ? 21. What fractions of a bushel will express 11, 15,25 and 32 quarts ? 22. What part of 8 is 5? 23. What parts of 11 are 2, 9, 12, 14, and 21 ? 24. What parts of 33 are 5, 7, 16, 25, and 32 ? 25. How many times is 15 contained in 34 ? 26. How many times, or, (more properly,) what part of a time, is 15 contained in 8 ? 27. What part of a time is 24 contained in 7 ? 28. What part of 16 is contained in 11 ? 29. What part of 12 does 5 contain ? 30. Divide 21 by 25 ; what is the quotient ? 31. If 17 be a dividend, and 25 the divisor, what must be the quotient ? 32. If 4 apples be divided among 5 boys, what part of an apple is each boy's share ? 33. If 3 men divide a barrel of apples equally among themselves, what fractions will express the shares of 1, 2, and 3 men ? 34. If 15 bushels of potatoes cost 7 dollars, what part of a dollar would 1 bushel cost ? 35. If 2 bushels of wheat sow 3 acres, what part of a bushel would sow 1 acre ? 36. If a cord of wood last 7 weeks, what part of a cord would last 1 week ? 37. Divide 16 by 17 ; what is the quotient ? 38. If 2 be a dividend, and 21 the divisor, what must be the quotient? 39. Divide 17 by 123. 40. Divide 84 by 1725. 41. Divide 1728 by 1837. 42. Express the division of 37 by 25. 43. Express the division of 25 by 36. 44. Express the division of 81 by 75. 45. Express the division of 16 by 9. 46. Divide 7 by 11. 89. Model of a Recitation. 1. If a man receive 125 dollars for J of his annual salary, what is his salary ? FRACTIONS. 71 Since J of anything make the whole of that thing, (83j) if \ of his salary is 125 dollars, |, or the whole of his salary, will be 4 times 125 dollars, equal to 500 dollars, which is the answer required. 2. 18 is :J of what number ? Since f of any number make the whole of that number, if one eighth oi some number is 18, the whole of that num- ber will be 8 times 18, equal to 144, which is the answer re quired. 9, Exercises in Finding the Whole of a Quantity FROM a single PaRT OF IT. In like manner^ solve and explain the following problems. 1. If ^ of a bushel of corn cost 42 cents, what is that a bushel ? 2. 42 is j- of what number ? 3. If -J of an acre produce 23 bushels of corn, how many bushels would 1 acre of land produce ? 4 23 is ^ oi what number ? 5. If ^ of the annual rent of a house be 75 dollars, how much is that for a year ? 6. 75 is ^ of what number ? 7. 25 is I of what number ? 8. 33 is ^ of what number ? 9. 16 is ^ of what number ? 10. If ^ of a mile is 40 rods, how many rods in a mile ? 11. If J of a- hogshead be 7 gallons, how many gallons in that hogshead ? 12. If ^^ of an acre is 4 square rods, how many square rods in an acre ? 13. If 60 minutes be ^V ^^ 3, day, how many minutes in a day ? 14. If 1 day be -r^-^ of a year, how many days in a year ? 15. At 35 dollars for working -^ of a year, how much is that for a year ? 16. If 25 cents make J- of a dollar, how many cents in a dollar ? 17. 62 is ^ of what number ? 18. 18 is 5^ of what number ? 19. John, being 12 years old, was only ^ as old as his grandfather. How old was John's grandfather ? 72 arithmetic. 91* Model of a Recitation. 1. John having 100 cents, paid away ^ of them for a pen- knife. How many cents did his penknife cost ? Since it takes four ^s of any thing, or numher, to make the whole of it, (SO,) if 100 be divided by 4, the quotient will be ^ of 100 cents, equal to 25 cents, which is the answer required. 2. What is i of 144 ? Since it takes | of 144 to make the whole of it, divide 144 by 8, and the quotient will be ^ of 144, equal to 18, which is the answer required. 93 Observation. Observe, that^ the dividerid being the proddci of the divisor and quotient^ C^,) the divisor shows how many equal parts ^ such as the quotient, (6S5) will make the dividend. Therefore, to ascertain any single part of a number, divide it by the numher which shows how many such parts make the integer, or given number, 93 Exercises in Finding a single Part of a Quan- tity from the Whole of it. In like manner, solve and explain the following problems, 1. If ^ of 100 cents be paid for a penknife, how many cents would the penknife cost ? 2. How many cents in ^ of a dollar ? 3. How many cents in | of a dollar ? in ^ of a dollar ? in |. of a dollar ? in ^ of a dollar ? in -^^ of a dollar ? in ^ of a dollar ? 4. If a ton of hay cost 21 dollars, what would if of a ton cost ? 5. What is \ of 63 ? of 72 ? of 81 ? of 90 ? of 99 ? of 108? 6. What is -^ of each of the following numbers : 11, 22, 33, 44, ^^, 99, and 132 ? 7. If a man, owning 279 acres of land, sell ^ of it ; how many acres would he sell ? 8. If 160 square rods make an acre, how many rods in \ of an acre ? 9. If 320 rods make a mile in distance, how many rods in i of a mile. FRACTIONS. 73 10. A furlong being ^ of a mile, how many rods in a fur- long ? 11. In a day there are 1440 minutes. How many minutes 12. In 'a pound there are 960 farthings. How many far- things in a shilling, which is ^V of a pound ? 13. If a slaughtered ox weigh 896 pounds, what would be th e weight of each quarter, the quarters being equal ? 14. A man hired a farm " at the halves," and raised 624 bushels of potatoes, 150 bushels of rye, 64 bushels of wheat, 7o bushels of oats, 12 bushels of white beans, 50 bushels of turnips, 25 bush, of corn, 45 bush, of winter apples, and 40 bushels of sauce apples. How many bushels in his share of this produce ? 15. If you could buy 480 apples for a dollar, how many could you buy up for ^ of a dollar ? for J- of a dollar ? for 4^ of a dollar ? for -j^ of a dollar ? . 16. If a man's salary be 800 dollars a year, how much is that for I of a year ? for J of a year ? for ^^ <^^ ^ 7^^ 17. If 32 quarts of nuts be divided equally among 4 boys, what part, and how much of them, is each boy's share ? 18. Divide 64 by 16 ; what part of 64 is the quotient ? 19. If you divide any number by 4, what part of that number will be the quotient? 20. What is ^ of 1000 dollars ? d4. Exercises in the Different Modes of Considering AND Reading Fractions. J of I is J, and | of 3 is three times as much, or | of 1 ; i of 5 is J of 1 ; 1 of 13 is Y" of 1 ; -J of 3 is f of 1 ; | of 5 is -f of 1. 1. How many ninths of 1 is ^ of 7 ? 2. J of 10 is how many thirds of 1 ? 3. i of 11 is what part of 1 ? 4. W hat part of 1 is ^^ of 25 ? 5. Read the following fractions in the four different modes described (84). h /t> a. h ih M. V- i^F^ T%^ ^^ ih ih M' V h ilh h h h A A V- 6. Which of these fractions expresses the greatest number of parts ? 7 74 ARITHMETIC. 7. Which expresses the largest parts ? 8. Which expresses the smallest parts ? 9. Which expresses the smallest number of parts ? 10. Which express the same number of parts ? 11. Which express parts of the same size ? 12. Which express just parts enough to make a unit? 13. Which express parts enough to make more than one unit? 14. Considering both the number and size of the parts, which is the largest fraction ? 15. Which is the smallest fraction ? 16. Why, of two fractions having equal denominators, is that greatest which has the greatest numerator ? 17. Why, of two fractions having equal numerators, is that greatest which has the smallest denominator ? 18. What effect is produced upon the value of a fraction by diminishing its numerator ? , 19. What effect is produced upon the value of a fraction by increasing its denominator ? 95# Expression, Definition, and Reduction of an Im- proper Fraction. As there is no limit to the number of parts that may be expressed by a fraction, (885) it is often convenient to ex- press in one fraction, more parts than there are of that size, in one tmit. But a fraction whose value is equal to^ or greater than its unit J is called an improper fraction ; and a fraction whose value is less than its unit, is called a proper fraction. The value of a fraction being the quotient resulting from the division of its numerator by its denominator, (865) an improper fraction may be reduced (208) to its equal inte- gral, or mixed number, by performing the division, which is only expressed by the fraction-. 96. Model of a Recitation. 1. A toll-gatherer took in one week ^-f-^ of a dollar, (four- pence-half-pennies ; ) how many dollars would they make ? Since there were 165 such parts of a dollar, that every 16 of them would make a dollar, (895) they would make as many dollars as there are times 16 in 165. Thus : -y^= 10^ dollars, which is the answer required. fractions. 75 9?. Exercises in the Reduction of Improper Fractions. In like manner^ solve and explain the following problems, 1. At a certain contribution, ^^^ of a dollar (ninepences) were taken ; how many dollars were taken ? 2. A merchant sold calico for ^ of a dollar a yard, till he re ceived -^^ of a dollar ; how many dollars did he receive ? 3. At a large party, ^^- of a pie were eaten, how many whole pies were eaten ? 4. In ^-f^- of a bushel how many bushels ? 5. In -^jV- of a pound how many pounds ? 6. In 4- ^^ ^ shilling how many shillings ? 7. In -^Z/- of a guinea how many guineas ? 8. In -^Jf ^ of a day how many days ? 9. In ^^%^ of an hour how many hours ? 10. In ^f f ^^ of a year how many years ? 11. Reduce -VV" ^^ units. 12. Reduce -ViV- to an integral number. 13. Reduce \^- to a mixed number. 14. Reduce ^^^- to a mixed number. 15. Reduce -yf- to an integral, or mixed number. 16. Change -*f f ^ to an integral, or mixed number. 17. Change |f| to an integral, or mixed number. 18. Reduce -fi^ to an integral, or mixed number. 19. How many units in Mj^-^? 20. What mixed number is equal to -If f ? 21. What is the value of |^ff in a mixed number? 08. Model of a Recitation. 1. Reduce 5^-^ to an improper fraction, that is, to six- teenths. Since there are 16 sixteenths in one unit, there will be 16 times as many sixteenths as units in any number. 16 times 5 are SO sixteenths y and the other 3 sixteenths -^^ 3 3 are ff, which is the answer re- ^ ^^' quired. S>9. Exercises in Reducing Integral and Mixed Num- bers TO Fractions.' In like manner, solve and explain the following probUTns, 1. Reduce 7 to sixteenths. 2. Reduce 25f to an improper fraction. 76 ARITHBIETIC. 3. Change ]2f to an improper fraction. 4. What fraction is equal to 3^^ ? 5. What is 10|- equal to in a fractional form ? 6. 13| are how many ninths ? 7. How many eighths of one dollar are 9 zi?AoZc dollars ? 8. How many Js of a yard are 32 yards ? 9. 1 5^1 days are how many ^^s of a day ? 10. 82^^ pounds are how many -^^ of a pound ? 11. 17|^ hours are equal to how many -^s of an hour? 12. 6^ hogsheads are equal to how many ^s of a hogshead ? 13. How many g-^-^s of a year are equal to 10 years ? 14. Keduce 437^y to an improper fraction. 15. Reduce lO^f-^- to an improper fraction. 16. Reduce 25^^^^ to an improper fraction. 17. What fraction is equal to 50-^ ? 18. Change 20 to sevenths. 19. Reduce 36 to twelfths. 20. Reduce 15 to fifths ; also to sixths. 21. Change 4 to halves, to thirds, to fourths, to fifths, and to sixths. 22. Reduce 16 to halves, to thirds, to fourths, and to fifths. 23. Reduce 1 to halves, to fifteenths, and to seventy-fifths. 24. Reduce 1 to halves, thirds, fourths, fifths, sixtfis, and to sevenths. 100 Model of a Recitation. 1. A man bought 25 yards of calico, at -3% of a dollar (3 fourpence-half-pennies) a yard ; how many dollars did his calico cost ? Since 1 yard cost ^^ of a dollar, 25 yards would cost 25 ^^=X5 = 4i , dollars. *T'' f ^f ^^2/ f ?^f ^^>^^ ^^ ^^ ^^ of a dollar, which are J-f of a dollar ; equal to 4^^ dollars, ( 95j ) the answer required. 101 Observation. Observe, (lOOj) that in multiplying the numerator ONLY by 25, retaining the same denominator , you multiply the fraction; for thus, you produce 25 times as many parts ( 83 ) of the same size. FRACTIONS. 77 103, Exercises in Multiplying a Fraction by an In- tegral Number. In like TTianner^ solve and explain the following problems, 1. How many dollars will 25 penknives come to, at f of a dollar apiece? 2. How many dollars would pay a man to work 5 days, at ^ of a dollar per day ? 3. How many dollars should Mr. Farmer receive for 12 bushels of corn, at f of a dollar a bushel ? 4. At y*2- of a dollar a pound for beef, how much would 1 1 pounds cost? 5. If a family consume | of a barrel of flour in a week, how much flour would last them a year? 6. If it take | of a bushel of rye to sow an acre, 15 acres would require how many bushels ? 7. If a horse eat ^ of a bushel of oats in a day, how much would keep him through December ? 8. If 1 bushel of apples cost ^ of a dollar, what would be the value of a load containing 33 bushels ? 9. At ^ of a dollar a day for board, what would be the cost of board for 365 days ? 10. How far can I ride in 1 hour at the rate of -^ of a mile per minute ? 11. How much is 5 times -J-^ ? 12. Multiply /^ by 13. 13. Multiply tJ^ by 43. 14. Multiply VbV by 36. 15. Multiply f ^ by 3. 16. How much is 15 times ^|- ? 17. Multiply ^^^ by 366. 18. How much is 3 times x^Jg- ? 103. Model of a Recitation. 1. At 32f dollars apiece, what would 7 cows cost? 095 Since 1 cow cost 32|^ dollars, 7 cows 17^ would cost 7 times 32f . Seven times f are ^, equal to f which 2283 dolls write, and 4 units, which add with the V ^ ' units, &c. (27) 2. How much is 83 times 16f feet ? It will be most convenient, in this example, to multiply the 7# 78 ARITHMETIC. integral and fractional parts separately, and add the products together. Thus : 16 48 ?^-a^ = J4& = 55^ feet. 128 1383^ feet. 104. Exercises in Multiplying a Mixed Number by AN Integral Number. In like manner, solve and explain the following problems, 1. If 15 yards are sufficient for one coat, how many yards will be sufficient for 10 coats ? 2. How many feet in 25 rods, there being 16^ feet in 1 rod ? 3. How many yards in 40 rods, there being 5^ yards in Irod? 4. How many cents in 6 shillings, there being 16f cents in 1 shilling? 5. How old is John, if he is 3 times as old as Charles, and Charles is 3^V5 years old ? 6. What would be the cost of 15 barrels of flour, at 6| dollars per barrel ? 7. If 31^ gallons make a barrel, how many gallons in 50 barrels ? 8. What is the price of a dozen bibles at 2f dollars apiece ? ^ 9. What is the cost of 10 dozen pairs of shoes at 1| dollars a pair ? 10. What would 7 tons of Lehigh coal cost at 9^ dollars a ton? 11. What would 17 grind-stones come to at 3-i^ dollars apiece ? 12. Multiply 6|-^ by 35. 13. How much is 100 times 2^ ? 14. What is the product of lf| multiplied by 5 ? 15. How much is 16f X 10 ? 16. Multiply 1728^^^^ by 7. 10l Model of a Recitation. 1. At -^^ of a dollar (3 fourpence-half-pennies) apiece, what would be the postage of 4 letters ? FRACTIONS. 7& Since the postage of 1 letter is -^^ of a dollar, the postage oi 4 letters would be 4 times as much. This product can be ascertained, either by multiplying the niTnerator by 4, retaining the same denominator ; (lOlj) or, far better, by dividing the denominator by 4, retaining the seme numerator. For, by the former process, you make the number of parts 4 times as large, the parts retaining the same size ; and, by the latter process, you make the size of the parts 4 times as krge, retaining the same number of parts. It is evident that, by the latter process, the parts are made 4 times as large, from the. fact that, it will take only J as 71 any of them to make the unit as before, -ff of a dollar are 12 fourpence-half-pennies, and J of a dollar are also 12 fourpence- 3 XA_ 1 2 of a dollar half-pennies ; for ^ of a dollar ^^ *^ ' is equal to 4 fourpence-half- ^3^^ = I of a dollar. f "^,^^' ^""^ * "^ "" dollar will ^^-* ^ be 3 times as majiy, or 12 four- pence-half-pennies. The two processes giving the same result, the latter is to be adopte^ in all cases when the multiplier is a factor (25) of the denominator ; because it will give the result in lower terms, |- being in lower terms, and, consequently, a more simple fraction than its equal -^f. 2. At f of a dollar a bushel, what would be the price of 8 bushels of potatoes ? Since the price of 1 bushel is f of a dollar, the price of 8 bushels would be 8 times as f q:^ = f = 3 dollars. much, which is 3 dollars, the answer required. For, by dividing the denominator by 8, the parts become 8 times as large, and such that each one of them makes a unit. 106. Observation. Observe, (lOS^) that by whatever number the denomina- tor IS DIVIDED, retaining the same numerator, the fraction IS THUS multiplied BY THAT NUMBER ; ^r the denominator shoioing the number of parts that make a unit, their size is INCREASED IN THE 'SAME RATIO .THAT THE DENOMINATOR IS DIMINISHED. Observe, also, that if a fraction be multiplied by its denomi- viator, the product will be the numerator. 80 ARITHMETIG. 107* Exercises in Multiplying a Fraction by dividing ITS Denominator. In like manner, solve and explain the following problems. 1. If 1 yard of calico cost |^ of a dollar, what would be the cost of 2 yards ? What would be the cost of 4 yards ? 2. At J of a dollar a pound, what will 2 pounds of butter cost ? What will 3 pounds cost ? What will 6 pounds cost ? 3. At |- of a dollar apiece, what would be the postage of 2 letters ? of 4 letters ? of 8 letters ? 4. If 1 ninepence is J of a dollar, what part of a dollar is 2 ninepences ? is 4 ninepences ? is 8 ninepences ? 5. If 1 fourpence-half-penny is -^^ of a dollar, what part of a dollar is 2 fourpence-half-pennies ? is 4 fourpence-half- pennies ? is 8 fourpence-half-pennies ? is 16 fourpence-half- pennies ? 6. If a sixpence is -^ of a dollar, what part of a dollar is 2 sixpences ? is 3 sixpences ? is 4 sixpences ? is 6 sixpences ? is 12 sixpences*? 7. At |- of a yard apiece for vests, how much satin would be necessary for 3 vests ? for 9 vests ? 8. At f of a mile a minute, how far would a train of cars run in 2 minutes ? in 3 minutes ? in 4 minutes ? in 5 minutes ? in 6 minutes ? in 10 minutes ? in 12 minutes ? in 15 minutes ? in 20 minutes ? in 1 hour ? 9. If it take l^f yards of broad cloth to make a coat, how much would it take for 3 coats ? for 6 coats ? for 8 coats ? for 12 coats ? for 24 coats ? 10. Multiply ^ by 5. 11. Multiply ft by 7. 12. Muhiply -J^^ by 25. 13. Multiply ^5% by 100. 14. Multiply II by 16. 15. Multiply 7| by 4. 16. Multiply 4^11 by 365. 17. How much is 20 times 9^ ? 18. How much is 327 times 10^ ? 19. J/- is -1 of what number ? 20. 5f is ^ of what number ? 21. 125^3^ is yV o^ what number ? 22. ^- is I of what number ? 23. Multiply -^-^ by 5, and that product by 3 it FRACTIONS. 81 ?H24. Multiply -^ by 5, and that product by 5. 25. Multiply j\ by 3, and that product by 5. 1.08. Model of a Recitation. 1. Multiply 5^ by 36. Since 36 is not a factor of the denominator, but 9, one of the factors of 3.6, is also a factor of the ^^^4 = JJ.= 3|. denominator, multiply first by 9, (^O,) by making the parts 9 times as large, (IO65) and then multiply that product by 4, the other factor of 36, by making 4 times as many parts, (IOI5) which will give 4 times 9 times, or 36 times ^, equal to -^5^, equal to 2^, which is the product required. 1.09* Exercises in Multiplying a Fraction by the Factors of the Multiplier. In like manner, solve and explain the following problems, 1. Multiply ^ by 18. 2. Multiply 4^ by 35. 3. Multiply II by 48. 4. How much is 24 times ^? * 5. How much is 50 times f f ? 6. How much is 81 times ^ ? 110 Model of a Recitation. 1. If 3 yards of calico cost ^^ of a dollar, (9 fourpence- half-pennies,) what would be the price of 1 yard ? Since 1 yard is -J of 3 yards, the price of 1 yard must be ^ of the price of 3 yards. ^^=-2.=^ of a dollar. Therefore, as the price of 3 yards is -^^ of a dollar, the price of 1 yard will be ^ (92) as Tnany sixteenths of a dollar, equal to y^^, the answer required. 111. Observation. Observe that, m dividing the numerator only by 3, retain- ing the same denomiTudor, you divide the fraction ; for thus you obtain \ as many parts (83) of the same size. 112. Model of a Recitation. 1. At 6 dollars a barrel, how many barrels of flour may be bought for 45^^ dollars ? 83 ARITHMETIC. Since 1 barrel costs 6 dollars, you may buy ^ (09) as many barrels as you have dollars ; -J 6 ) 45i\ of 42 is 7. Reduce the remaining 3 to elevenths making 33, these 7^ barrels. and the other 3 elevenths are ff , ^ of which are ^y of a barrel, which written with the 7 barrels make 7-j-\ barrels, the answer required. 1 13. Exercises in Dividing a Fraction by an Integral Number. In like manner, solve and explain the following problems. 1. If 2 bushels of potatoes cost f of a dollar, what would that be a bushel ? 2. If a cow consume |- of a bushel of meal in 3 days, how much would that be per day ? 3. At 1% of a dollar for 4 pounds of beef, what would be the cost of 1 pound ? 4. If 4 horses consume ^f of a ton of hay in a month, how much would that be for 1 horse ? 5. At 14- of a dollar for 7 pounds of coffee, what would be the cost of 1 pound ? 6. If 2 yards of cloth cost 8| dollars, wha< would 1 yard cost at that rate ? 7. What would be the cost of 1 bushel of wheat, if 4 bushels cost 32|- shillings ? 8. If I give 23| bushels of wheat for 3 sheep, how much would that be apiece ? 9. If I give 59^ bushels of corn for 7 calves, how many bushels would that be apiece ? 10. If 20|- dollars be paid for 15 days' work, how much would that be per day ? 11. How far per hour is 88^ miles in 17 hours? 12. How far per day is 476f miles' travel in 8 days ? 13. If 15 men divide among themselves 77^ barrels of apples, what would be the share of each man ? 14. If 23 yards of cloth cost 152f dollars, what would that be a yard ? 15. How much is the cost of 1 yard of cotton cloth, when 3jVtt dollars are given for 35 yards ? 16. How many times is 25 contained in 59f ? 17. What is -fV (93) of 148^? 18. Divide 5| by 12? FRACTIONS. 83 19. How many times is 9 contained. in 47 ^^ ? 20. What is i of H? 21. Whatis^of2|? 22. What is i of 18f ? 23. Divide 4f by 5. 24. Divide 1731f by 12. 25. Divide 65542^^ by 256. |j|j26. Divide 16388^ by 128. Ul4. Model of a Recitation. 1. If the postage of 4 letters, between the same towns, be I of a dollar, how much would that be apiece ? Since 1 letter is J of 4 letters, the postage of 1 letter must be J of the postage of 4 letters. |_-^ =-j?^ of a dollar. Therefore, as the postage of 4 letters is | of a dollar, the postage of 1 letter would be J of | of a dollar. But the nurri' her of parts not having 4 for a factor, you must perform the division upon the size of the parts, which you can do by multiplying thP denominator by the divisor, retaining the same numerator. It is evident that, by multiplying the denjominator by 4, the parts are made \ as large, from the fact that, it will take 4 times as many of them to make the unit as before. It takes only ^fourths of a dollar (4 quarters of a dollar) to make a dollar ; whereas it takes 4 times as many, or 16 sixteenths of a dollar, (16 fourpence-half- pennies,) to make a dollar. 115>, Observation. Observe, (ll^j) that by whatsoever number the denomi- nator of a fraction be multiplied, retaining the savne numerator, the fraction is thus divided by that number ; for the denominator showing the number of parts that make the unit, (83^) their size is diminished in the same ratio that THE denominator IS INCREASED. Observe, also, that the division of a fraction may be per- formed, either upon the number of the par^fs, their size remain- ing the same, (IIO5) or upon the. size of the parts, their number remaining the same, (1 14: 5) hut, that the former pro- cess is to be adopted in all cases ivhen the divisor is a factor of the numerator^ because it will give the result i7i lower terms. I 84 arithmetic. 116. Exercises in Dividing a Fraction by Multiply- ing ITS Denominator. In like manner^ solve and explain the following problems. 1. If 2 boys having J of a melon divide it equally between themselves, what would be the share of each ? 2. What is i of I ? 3. Suppose I of a pie to be cut into 2 equal pieces, what part of the whole pie would each piece be? What is \ ofi? 4. A boy having f of a dollar, gave \ of it for a pen- knife. What part of a dollar did his knife cost ? What is \ of I ? 5. If 2 shillings are J of ar dollar, what part of a dollar is 1 shilling? 6. If a boy having -5^ of a pie should give \ of it to his sister, what part of a pie would he give away, and what part would he keep ? 7. If ^ of a dollar be divided equally among 3 boys, what part of a dollar is the share of each ? 8. If you should make a circle on your slate, and draw a line across it through the centre, how many parts would you make of it ? What would be the name of each part ? 9. If from the centre of said circle you draw a line through the middle of one half, making two parts of that half, how many such parts would make the whole circle ? What is the name for such parts ? What is ^ of ^ ? 10. If from the centre of said circle you draw a line through the middle of one fourth, making two parts of that fourth, how many such parts would make the whole circle ? What is the name for such parts? What is 4 of J ? 11. What is 1 of ^ ? W^hat is | of yV ? 12. If 3 pounds of butter cost of a dollar, what is that a pound ? 13. At I cf a dollar for 4 bushels of apples, what would be the cost of a bushel ? 14. At I of a dollar for 7 gallons of vinegar, what would be the cost of a gallon ? 15. If 6 bushels of wheat cost 4|- dollars, what would that be a bushel ? 16. If 4 dollars buy 5 bushels of rye, how much would one dollar buy ? ^K FRACTIONS. 85 i % If 4 dollars buy 3J yards of silk, how much might be ught for 1 dollar ? 18. If 18 pounds of raisins cost 2f dollars, what is that a pound? 19. If 16 hats cost 48| dollars, what would 1 hat cost ? 20. What would 1 yard of broadcloth cost, if 25 yards ccst 150| dollars? 21. How far per hour would a train of cars go, if it run l'!5J miles in 7 hours ? 22. Divide 24| by 7. 23. What part (87) of 5 is 2? 24. What part of 5 is 2i ? 25. What part of 5 is l| ? 26. What part of 12 is 7 ? '27. What part of 12 is 3^\ ? 117. Illustration of the Principle of Dividing by the Factors of the Divisor. By multiplying one number by another, we introduce ipto the multiplicand all the factors composing the multiplier, (Sdj) and the product will be composed of all the factors of both multiplier and multiplicand. Also, in dividing one number by another, we take from the dividend all the factors composing the divisor, (755) and the quotient will be com- posed of all those factors composing the dividend, which are not necessary to compose the divisor. Thus, by multiplying 35 = 7 X 5 by 33 = 3 X H, vve obtain 1155 = 7 X 5 X 3 X 11. Now, by dividing 1155 = 7 X 5 X 3 X H by 105 = 7 X 5 X 3, the quotient is the remaining factor, 11 ; or, if we divi4^-by 35 = 7 X 5 the quotient is 33 = 3 X H. the remaining two factors ; or, if we divide by 7, the quotient is 165 = 5 X 3 X 11) the product of the remaining three factors. Consequently, by dividing the product of several factors by some of them, the quotient will be the product of the others. Also, when convenient, we may separate a divisor into factors, and take them from the dividend, one at a time, that is., divide first by one factor, then divide the quotient, thus obtained, by another factor, and so on, wfth all the factors of the divisor ; the last .quotient will be the quotient required. Thus, instead of dividing 1155 directly by 21, we may divide first by 7, obtaining 165, which divided by 3 gives 55, the true quotient. 8 96 arithmetic. 118. Model of a Recitation. 1. Divide 875 by 35. 7 ) 875 '^^^ quotient is ^\ of the dividend. Divide ' first by 7, one of the factors of the divisor, to 5 ) 125 obtain j- of the dividend, and then divide the ' quotient thus obtained, by 5, the other factor 2^ of the divisor, to obtain -J of -f , or -^ of the dividend, as required. 2. Divide 3| by 36. Since 36 is not a factor of the numerator, but 4, one of the factors of 36, is also a factor of the numer- 3^ = -^. ator, divide first by 4, by taking J as many parts, (lllj) and then divide that quotient ^5^~5^" = ^?- % ^' ^^^ other factor of 36, by making the parts ^ as large, (114,) which will give i of J, or -rf^ of -^, equal to ^, which is the true quotient required. 119. Exercises in Dividing by the Factors of the Divisor. hi like manner, solve and explain the following problems, 1. Divide 1421 by 49. 2. How many times is 72 contained in 1728 ? 3. How many casks of 63 gallons each, may be filled from 7875 gallons ? 4. If a horse travel f of a mile in 12 minutes, how far would he travel per minute ? 5. If 21 dollars buy 3^ barrels of flour, what part of a barrel would 1 dollar buy ? 6. How manv times is 14 contained in 72| ? 7. Divide 12| by 15. 8. Divide 108f by 18. 9. How many times is 30 contained in 72| ? 10. What part of a time is 27 contained in 3^^? 11. What part of a time is 28 contained in 8| ? 12. How many times is 36 contained in 42-^? 13. Divide 175 by 21. 14. Divide 1836 by 24. 15. What is the quotient of 960 divided by 45 ^ 16. Divide 2^-2^ by 24. 17. Divide 38f^ by 36. FRACTIONS. 87 18. Divide 3492 by 81. 19. What part of 15 is 10 ? 20. What part of 18 is 5f ? 21. What part of 21 is 4J ? 22. If 8512 be the product of three factors, two of which aie 8 and 19, what is the third factor ? 23. If 17160 be the product of 8, 11, 13, and two other factors, what are the other two factors ? 24. One of two factors composing 1625 is 25. What is the other ? 25. Divide 17 X 19 X 10 by 19. 26. Divide 12 X 14 X 9 X 6 by 12 X 6. 27. How many times is 3x5x7 contained in 9 X 10 >< 14? 28. What is the quotient of 16 X 39 X i divided by 2 X 7X 13? 130. Illustration of the Principle of Reducing a Frac- tion TO Other Terms of Equal Value. Make a circle on your slate, and draw a line across it through its centre, making two half-circles. From the centre draw a line through the middle of one of these halves, and from the same point draw a line through the middle of each of the two fourths made of this half by the last line ; thus making the ^ = f . Now erase the three lines last drawn ; thus making the f = | again. 21. Observation. Observe, (120^) that both terms in \ are made 4 tiTues as large in its equal fraction | : that is^ both terras in J have been multiplied by 4 ; thus making the parts 4 times as many^ but \ as large in ^ as in ^, ^ jSo, multiplying both terms of a fraction by any number, will reduce it to an equal fraction in higher terms, For^ tohile it multiplies the fractio?i by increasing the number oj parts, (lOlj) it also divides it by diminishing the size 86 ARITHMETIC. of the parts (114) in the same ratio as their nwmber is in^ creased. Observe, also^ that both terms in | are made J as large in its eqvul fraction ^ ; that is, both terms in f have been divided by 4 : thus making the parts \ as many, but 4 times as large in ^ as in %, So, dividing both terms of a fraction by any number, will reduce it to an equal fraction in lower terms. For, while it divides the fraction, by diminishing the number of parts, (11 Ij) it also multiplies it by increasing the size of the PARTS (106) in the same ratio that their number is dimin^ ished. 133. Illustration of the Mode of Reducing a Frac- tion TO Lower Terms. 1. Reduce f J to lower terms. Since 3, 7, and 21 are factors com- ^) ii = -^ f'^^^on to both terms of the fraction, 7\2i-_.3\3 1 you may divide both terms (131) /ft ) r^ t ^y either of these common factors, ^1 ) f i = i But observe, that, the larger the factor used, the lower will be the terms to which the fraction will be reduced ; and that, by using the greatest common factor , the fraction will be reduced to its lowest terms. When any other than the greatest common factor is used, the new fraction obtained may be reduced lower, by using some other common factor. 133. Factoring of Numbers. In small numbers, the factors and common factors may be ascertained by observation ; but in larger numbers other means become necessary. A multiple of a number is a number of which the former number is a factor; as multiples of 3 are 6, 9, &c., of which 3 is a factor. A common multiple, of two or more numbers, is a number of which those numbers are factors. A number composed of factors is a multiple of any one of those factors ; and, also, of any combination of its prime fac- tors. (39.) When one number is a factor of another, all the factors of the former are also factors of the latter. Thus, 21 being a factor of 63, 7 and 3, the factors of 21, are also factors of 63. FRACTIONS. 89 And they should be ; for 63, being 3 times 21, should contain 7 and 3 three times as often as 21 contains them. Hence, a factor of a number is a factor of any multiple of t lat number. Any common factor of two numbers is a factor of their sum, and of their difference. For, each of the numbers con- taining the common factor a certain number of times, their sum must contain it as many times as both of the numbers ; End their difference must contain it as many times as the krger of the numbers contains it times more than the smaller. Thus, 4, being a common factor of 12 and 20, must be a factor of their sum, and of their difference : for 12 is 3 fours, E.nd 20 is 5 fours ; their sum will be 5 fours -j- ^ fours = 8 fours, or 32 ; and their difference will be, 5 fours 3 fours == 2 fours, or 8. 2 is a factor of every even number. Any number ending with a cipher (10) is a muUiple oi ".0, consequently, 10, and the factors of 10, are factors of it. Any number ending with two ciphers (10) is a multiple of .00, consequently, 100, and the factors of 100, are factors of it. 5 is a factor of any number ending with 5 ; for all of the number but 5 is a multiple of 10 of which 5 is a factor. Any factors of the last two figures of a number, which are also factors of 100, are factors of the whole number ; for all of the number but these two figures is a multiply of 100. 8, being a factor of 200, will be a factor of any number which has even hundreds, if it be a factor of the last two figures of the number. 9 is a factor of any number, when it is a factor of the sum of the digits which express that number. For the excess in the value expressed by the digits, above what they would ex- press in the units' place, is a multiple of 9 ; since every re- moval of a figure one degree higher causes that figure to express teii times its former value, (4,) it gains by each removal 9 times the value it would have before the removal. Thus, 10 is 9 more than 1 ; and 100 is 9 X 10 = 90 more than 10, or 99 more than 1, &c. Also, 70 is 9 X 7 = 63 more than 7 ; and 700 is 9 X 70 = 630 more than 70, or 9 X 70 + 9 X 7 = 693 more than 7. Also 3, being a factor of any multiple of 9, is a factor of any number when it is a factor of the sum of the digits which ei^press that number. 90 ARITHMETIC. Every factor of a number has its corresponding factor, which, together, compose the number, (75.) Hence, to find the prime factors of a number, separate it into two factors, by dividing it by any known factor, and proceed in the same manner with each of these factors, and so on, till the prime factors are all obtained. 134. Model of a Recitation. Find the prime factors of 1296. Here observe^ that 12 is a factor, the factors of which are 3, 2, 2. The quotient of 1296 divided by 12 is lOS, whose factors are 12, the factors of which are 3, 2, 2 ; and 9, the factors of which are 3, 3. In all, 3, 2, 2 X 3, 2, 2 X 3, 3 ; or, 3, 3, 3, 3, 2, 2, 2, 2 ; or 3^ 2\ the small 4 being an index to show the number of factors like that over which it is placed, (364.) 1S5, Exercises in Factoring Numbers. In like manirher^ solve and explain the following problems. Find the prime factors of the following numbers. 72, 88, 120, 612, 336, 648, 930, 924, 936, 450, 360, 966, 870, 684, 396, 432, 2480, 8000, 10449, 10503, 24876. 136. Model of a Recitatioti. Reduce %% to an equal fraction in its lowest terms. Take -^t^ as many parts, (lll^) and make 12 ) 14 = ! them 12 times as large, ( 106^} which gives f , the answer required. 1S7. Exercises in Reducing Fractions to Lower Terms. In like manner^ solve and explain the following problems, 1. Reduce -^ to its lowest terms. 2. Reduce -/_, |2^ ^4^ ^a, J5^ i| ^nd ^| to their lowest terms. 3. Reduce J|, ^^-, f |, f|, -^^j, and -f^ to their lowest terms. 4. Reduce -^^j, f |^, -^i^-^ and Jfg to their lowest terms. 5. Reduce /^o^, J^Vf^ Hf ^nd yVV? to their lowest terms. 6. Reduce yVsV ^^ ^^^ lowest terms. FRACTIONS. 91 128 Model of a Recitation. It is often most convenient, in reducing a fraction to its lowest terms, to make use of the greatest common factor of its terms, (122.) Hence it will be useful to find a more d rect way by which the greatest common factor may always b'i ascertained. Reduce ^^^ to its lowest terms. The greatest common factor, being a 61) 14S (2 factor of 148 and of 64, consequently, 128 (123.) of 64 X 2= 128, will be a factor of 148 128 = 20, (123.) 20)64 (3 Again, this factor, being a factor of 60 ^ 64 and of 20, consequently, (123,) of 20 X 3 = 60, will be a factor of 4)20(5 64 60 = 4, (123,) But 4, being 20 a factor of 20 and of itself, (123,) will be a factor of 20 X 3 + 4 = 64, - g^ ig one of the given numbers. Again, 4, ^ ) Tf B Tt being a factor of 64 and of 20, will be a factor (123) of 64 X 2 + 20 == 148, the other given number. Hence, as 4 contains the greatest common factor of the given numbers, and is a factor of therny it must be their greatest common factor. Therefore, take \ as many parts, and make them 4 times as large, which gives -Jf , the answer required. 129. Observation. Observe, (128^) that^ the greater of two given numbers being divided by the less, the less by the first remainder, the first remainder by the second, the second by the third, <^c,, till there be no remainder ; the greatest common factor of the given numbers will be a factor of the several remainders ; for the remainders are differences (123) between numbers of lohich this greatest common factor is a factor. Consequently, the greatest common factor of the given numbers cannot exceed the last remainder. But THE LAST REMAINDER IS ITSELF THAT FACTOR ; for, retracing the several remainders and given numbers from the lost remainder to the larger given number, observe tJiat the last remainder is a factor of the next precede 92 ARITHMETIC. ing ; that each of them ^ added to the next preceding, or a multiple of it, makes the next in order ; and that, therefore, the lojSt remainder must he a factor of them all, (I$J3 2) hence, as the last remaindei , both contains the greatest common FACTOR of the given numbers, and is a factor of them, it must he their greatest common factor. 130* Exercises in reducing Fractions to their Lowest Terms by the Greatest Common Factor. In like manner, solve and explain the folloioing prohlems, 1. Ascertain the greatest common factor of 30 and 72. 2. Reduce ^^ to its lowest terms. 3. Ascertain the greatest common factor of 126 and 342. 4. Reduce -Jj-|- to its lowest terms 5. Ascertain the greatest common factor of 128 and 176. 6. Reduce ^f to its lowest terms. 7. Reduce -ff f to its lowest terms. 8. Reduce ^/^ to its lowest terms. 9. Reduce ^^|- to its lowest terms. 10. What is the greatest common factor of 384 and 1152 ? 11. What are the lowest terms of ^^^? 12. What is the greatest common factor of 114 and 285 ? 13. Reduce ^^^ to its lowest terms. 14. Reduce ^f f f to its lowest terms. 15. What are the lowest terms of 7^ ? 16. What are the lowest terms of ylf^ ? 17. Reduce -^-f^is to its lowest terms. 131. Illustration of the Least Common Multiple op Numbers. 2 is a factor of 4, 6, 8, 10, 12, 14, 16, 18, &c., and 3 is a factor of 6, 9, 12, 15, 18, &c. ; consequently, 4, 6, 8, 10, &c., are multiples of 2; and 6, 9, 12, &c., are multiples of 3. But 6, 12, 18, &c., are mul- tiples of hoth 2 and 3 ; hence, they are common multiples of 2 and 3 ; and 6 is the least common multiple of 2 and 3. 133. Illustration of the Least Common Denominator OF Fractions. 1. Reduce \, also \, to equal fractions in higher terms. I ^^1 FRACTIONS. 93 multiplying both terms of each fraction by 2, 3, 4, 5, kc., successively, > becomes < 4 ) ( f == f = A- = A = f?> &c. Observe, that the denominators of the fractions to which \ may be reduced, will be multiples of 2, the denjominator of \ ; and that the denominators of the fractiAms to which \ may he reduced^ will be multiples of^, the denominator of\. But, PARTICULARLY OBSERVE, that the COMMON MULTIPLES of 2 and 3, the denominators of \ and \, may be common de- nominators of fractions to which \ and \ may be reduced ; %nd that the least common multiple of the denominators of\ 2nd \ will be the least common denominator to which I and J can be reduced, 133. Model of a Recitation. 2. Reduce f and J to equal fractions having their least common denominator. By multiplying both terms of |=:^ = -5;f = f J each fraction by 2, 3, 4, &c., suc- cessively, you obtain for denomina- f = -^^ = j^ tors all the multiples of the given denominators as far as you proceed ; consequently, the first common denominator thus obtained, will be the least common denominator of the given fractions. I134L* Exercises in Reducing Fractions to their Least ^ Common Denominator. In like manner, solve and explain the following problems. 1. Reduce f and | to equal fractions having their least common denominator 2. Reduce \ and f to their least common denominator. 3. Reduce % and \ to their least common denominator. ' 4. Reduce \ and f- to their least common denominator. 5. Reduce f and f to their least common denominator. 6. Reduce | and f to their least coihmon denominator. 7. . Reduce | and f to their least common denominator. 8. Reduce f and I- to their least common denominator. 9. Reduce \ and -^^j to their least common denominator. 94 ARITHMETIC. 10. Keduce f and -^^ to their least common denominator. 11. Reduce ^jj and /g- to their least common denominator. 12. Reduce ^'^ and -f^ to their least common denominator. 13. Reduce -^ and f^ to their least common denominator. 14. Reduce -fg and fy to their least common denominator. 13S Mode of finding the Least Common Multiple. If you know the right numbers by which to multiply both terms of each fraction, to reduce the fractions to their least common denominator, only one multiplication for each fraction would be necessary. Hence, as you will often have occasion to reduce fractions to their least common denominator, it is desirable to find a more direct way to ascertain the right multipliers. Every number which is not a prime number, is composed of prime factors, (29.) Thus : 24 = 3x2x2x2. Though 4, 6, 8 and 12 are factors of 24, yet they them- selves are composed of prime factors, and, therefore, are com- posite factors, A multiple, or composite number, is composed of exactly all its prime factors. Hence, a number which contains the prime factors of another number, is a multiple of that other number ; also, a number which contains the prime factors of two, or more other numbers, is a common multiple of those other numbers. Consequently, the least common multiple of two or more giv7inumbers,will be composed of suck of their prime factors, and only such, as are necessary to compose each of the given numbers. Thus: 6 = 3x2, and8=:2x2x2; now take 3x2, the factors of 6, and 2x2, the factors which 8 has that 6 has not, and you have all the factors of 6 and 8 ; viz : 3 X 2 X 2 X 2=: 24, the leasi common multiple of 6 and 8. Hence, to ascertain the least comrnon multiple of two or more given numbers, it is only necessary to separate the given numbers into their prime factors, and to select and multiply together such, and only such of the factors as are necessary to compose each of the given numbers, 130* Model of a Recitation. 1. Ascertain the least common multiple of 14 and 21. I FRACTIONS. 95 Ij^ o v7 ^^ iskes 2 and 7 to compose 14, 21 ZI Q V 7' ^^^ ^^^^ *^^^ '''' ^^o^^^^ with the sT^ V ?9 ' 40 3, compose 21 ; therefore, the other ^X ^ X cJ ^. ^ ^^.^^ omitted, 2 X 7 X 3 = 42, 'vrill be the least common multiple required. ]l37* Mode of reducing Fractions to their Least Com- mon Denominator. * 2. Reduce -^^ and-^T to equal fractions having their least common denominator. Since the least common denominator will be the least com- mon multiple of the given Ti = ^ = YKTh='T^' denominators, (ISS,) it will only be necessary to ^ = ^ = ^f^2- = :fV separate the given denom- inators into their prime factors, and multiply both terms of each fraction by such factors composing the denominator of the other fraction, as are necessary to make each denominator equal to the least common multiple of the given denominators ; that is, multi- ply both terms of each fraction by the factors, composing the denominator of the other fraction, which it has not already in its own denominator. Thus ; by multiplying both terms of the first fraction by 3, and of the second by 2, the denominators will be com- posed of the same factors, and only such as are indispensa- ble ; consequently, the fractions are reduced to equal fractions having their least common denominator as required. 138. Model of a Recitation. 3. Reduce ^ and ^^ to equal fractions having their least common denominator. First, reduce the frS.ctions to ^^ = I = ^1^ = ^^.. their lowest terms, then separate the denominators into their prime -J-f == yV == :f 1; 3- = i-J- factors, or,^ since they have a common composite factor, 4, this need not be reduced to prime factors ; and, finally, multiply both terms of the first fraction by 3, and of the second by 2f and the fraction will be reduced as required. 96 ARITHMETIC. 4. Reduce ^, ^, and ^5-, to their least common denomi- nator. Multiply both terms of the first |- = ^^Tj = ^. fraction by 5, of the second by 3, ^ 3 g and of the third by 2, and the several 1^ ^x's UTF* denominators will be composed of -^^ = -j4-^ = -^jj, the same factors ; consequently, the given fractions will be reduced to their least common denominator as required. , 1.39. Observation. Observe, that, to reduce two or more fractions to their least common denominator^ we first reduce the fractions to their loioest terms ; second, separate these denominators into their prime factors ; and third, multiply both terms of each of these fractions by the factors belonging to the other de^ nominators lohich do not belong to its own denominator. 140 Exercises in reducing Fractions to their Least Common Denominator. In like manner, solve and explain the following problems, 1. Ascertain the least common multiple of 8 and 12. 2. Reduce f and y^^ to their least common denominator. 3. Ascertain the least common multiple of 8 and 14. 4. Reduce f and -f-^ to their least common denominator. 5. Ascertain the least common multiple of 9 and 15. 6. Reduce f and -^-^ to their least common denominator. 7. Ascertain the least common multiple of 15 and 18. 8. Reduce -f-^ and -f^ to their least common denomina- tor. 9. Ascertain the least common multiple of 5 and 7. 10. Reduce f and f to their least common denominator. 11. Ascertain the least common multiple of 2, 3, 5, and 7. 12. Reduce \, \, \, and f, to their least common denomi- nator. 13. Ascertain the least common multiple of 10, 14, and 15. 14. Reduce -^j, y\-, and -^ to their least common denomi- nator. 15. Ascertain the least common multiple of 250 and 400. w FRACTIONS. 97 Reduce -^^ and -^^ to their least common denomi- nator. 17. Ascertain the least common multiple of 15, 24 and '15. 18. Reduce ^, ^-j? Q^^^d,-^^, to their least common de- nominator. 19. Reduce J-f- and -f^ to their least common denominator. 20. Reduce -^ and ^^ to their least common denominator, 21. Reduce J^, f|, and f|, to their least common denom- iaator. 22. Reduce ^ and -^5- to their least common denominator. 23. Reduce y\, -^j, and ^^, to their least common de- r.ominator. 24. Reduce -1^5^(5- and yf fiy to their least common de- rominator. 1141 . Model of a Recitation. 1 . John paid f of a dollar ( 5 ninepences) for a reading book, J of a dollar for a writing book, and j of a dollar for an arithmetic ; how many dollars did they all cost ? Since the parts expressed by ^^^Z. = -1^3- = 1| dolls. these several fractions are all eighths^ and since the numerator of each fraction shows the number of parts expressed by that fraction, (83^) the sum of the numerators will show the number of parts expressed by all of the fractions ; therefore, place the sum of the numerators over their common denomi- nator, and the result will be the sum of the fractions, as required. 2. If I pay 2| dollars for a pair of shoes, and 4f dollars for a pair of boots, what is the whole cost ? Here are 3 parts and 5 parts making 2| = 2y^2- S parts, but they are all neither fourths^ 4^-^410 nor sixths ; if, however, you reduce the fractions to their least common de- 77 dollars nominator, ( I3O5) the parts become ^ A + = ^A^ = +I=1t\-. Write the ^3^, and add the unit with the other units, makmg T^^g dollars, which is the answer required. 98 ARITHMETIC. 143* Model of a Recitation. 1. A boy, having f of a dollar, spent f of a dollar for a bunch of quills. How much money had he left ? He had left the difference ^ g 3 I n 111 between f and f . Since the w^ ^ t parts expressed by the frac- tions are all sixths, and the numerators show the number of the parts, the difference be- tween the numerators will show the number of parts he had left, which continue to be of the same size ; therefore, place the difference of the numerators over the common denominator^ and the result will be the difference between the fractions, as required. 2. If a man earn 14| dollars, and spend 4|- dollars in a week, what would he save in a week ? He would save the difference j^i ___ 2^3 between what he earned and what .5 .5 he spent. % -^ % You cannot take 5 parts from 3 "~ parts of the same size ; therefore, 9^ dollars. reduce 1 of the 14 units to sixths, (OSj) making 6 sixths, and the 3 sixths, make |, from which, if f be taken, there Avill remain ^-^ == |, which write ; and then take 4 units, not from 14 units, for 1 of them has been disposed of, but take 4 units from 13 units, and there will remain 9 units ; making 9| dollars, which is the answer required. 14:3. Exercises in adding and subtracting Fractions. In like manner, solve and explain the folloiving problems. 1. If you buy a lead-pencil for -Jg- of a dollar, a writing- book for -f^ of a dollar, an inkstand for y\ of a dollar, how much must you pay for the whole ? 2. At a contribution, John contributed -^ of a dollar, his brother ^ of a dollar, and their sister -^^ of a dollar. What did they all contribute ? 3. By going in the road, John walks |- of a mile to school, but by going across the pastures and fields, it is only f of a mile to school. How much can he save in distance by going the nearer way ? 4. If a writing book cost J of a dollar, and a quire of letter FRACTIONS. 99 paper cost ^ of a dollar, how much more will the paper cost t lan the book ? 5. If Samuel have f of a dollar, and Martin have of a dollar, how much have both of them ? 6. If Isaac have | of a dollar, and his sister f , which has the more money, and how much more than the other ? 7. How many yards of cloth in 4 pieces which measure as follows, 18f yards, 27|- yards, 23| yards, and 25| yards ? 8. If Mr. Farmer hire 2 men and a boy to work for him a week, and pay them as follows, 5f dollars to one man, 7 1 dollars to the other man, and 3| to the boy ; how much v^ould he pay the whole ? 9. If it take 1 1 yards of cloth to make a coat, and | of a yard to make a pair of pantaloons, how much more cloth in tlie coat than in the pantaloons ? 10. A merchant bought a piece of cloth, containing 23 yards, and sold 7f yards of it. How much of it had he left ? 11. In a barrel there are 31 J gallons, and in a hogshead 63 gallons. How many more gallons in a hogshead than in a barrel ? 12. If 7 1 gallons leak out of a barrel, how much would remain ? 13. John works ^ of the time, plays J of the time, sleeps J of the time, and is at school the rest of the time. What part of the time is he at school ? 14. Of the road that John walks to school, J is up hill, J is down hill, and the rest is level. What part of the way is level road ; and how much more of the way is up hill in going to school than in returning home ? 15. A pair of oxen and a horse compose a team ; one ox draws f of the load, the other ox draws | of the load, and the horse draws the rest of it. How much more do the oxen draw than the horse ? 16. Add together f and ^^. 17. What is the sum, and difference of f and J ? 18. Add together 7| and lOf. 19. What is the difference between 13 5*5 and 17^ ? 20. What is the sum, and difference of 16^ and 12^\ ? 21. Subtract 24^3^ from 25t^. 22. How much more is 12^^^ than both 4f and 5^^ ? 23. How much less are both f and 2 j than 4 ? 24. How much more is the sum of 10/^ and 5^ than their difference ? 100 ARITHMETIC. 25. How much is 1^--^-^ ? 26. How much more is ?2 +|j:i2 than '1^^ ? 27. How much less is l^V'-^ than ^-t^fi^ ? 28. How much are i|--^5 and i^? ? 29. Add together ^,1, J, and i 30. Add together ^\, -if, f f , and f . 31. Subtract ^^\ from f f . 144: Illustration of the Principle of multiplying by A Fraction. At 4 dollars a yard for broad cloth, what would be the cost of 4 yards ? of 2 yards ? of 1 yard ? of J of a yard ? of I of a yard ? If 1 yard cost 4 dollars, 4 yards 4 X 4 = 16 dollars. would cost 4 times 4 dollars, equal A Ky o Q A u to 16 dollars. 2 yards would cost 4 X ^ dollars. 2 ^.^^^ ^ ^^^^^^^ ^^^^^ ^^ g ^^^^^^^^ 4x1= 4 dollars. 1 yard would cost 1 time 4 dollars, - 1 o /I 11 equal to 4 dollars. | of a yard ^^ ^^ ' which ascertain by making 3 times as many parts, (101.) For I of a day's work, he should receive ^ of f of a dollar ; which ascertain by taking | as many parts, (111.) For J of a day's work, he should receive J of f of a dollar ; which ascertain by making the parts J as large, (114.) 2. If a horse travel 6| miles per hour, how far would he travel in 4 hours ? in | of an hour ? in 5 J hours ? If he travel 6| = ^ miles in 1 ?1_ __ 27 miles. hour, in 4 hours he would travel ~ 4 times ^ of a mile ; which ascer- 27-^3 o Ai ^:Uq tain by making the parts 4 times 4^2 f ^2 ^i^es. ^^ ^^^^^^ (106.) In I of an hour 27-^3X4 oc -1 ^ would travel | of ^- of a mile. TT4 = ^^ ^^^' Divide the number of parts (111) by 3, to obtain J, and make these parts 2 times as large, (106^) to obtain |, which, reduced, will be the answer required. In 5J = J^ hours, he would travel -^t^^- of ^- of a mile. Divide the number of parts (111) by 3, to obtain |, and multiply the quotient by 16, to obtain ^ ; but, since 4, one of the factors of 16, is a factor of the denominator, (IO85) multiply by 4, by making the parts 4 times as large, and then multiply by 4 again, the other factor FRACTIONS. 103 ' 16, by making 4 times as many parts, which, reduced, will b3 the answer required. In reducing this expression of the a iswer, say : 3 in 27, 9 times, and 4 times 9 are 36. 4 in 4, oice ; and, since the denominator is 1, the numerator, 36, is units, (83.) Another Explanation. If he travel 6|, or ^/-, miles in 1 hour, in 5\ hours he would travel 5J, or -^, times as far. r irst, multiply by 16, as if it were units, which gives ^i^/ ; bit, as the right multiplier is J^, only \ of 16 units,* the right product ought to be only \ of what we now have ; there- f(tre, divide by 3, which gives Y-V " ^ = 36, as before. 149. Observation. Observe, (148,) that^ in multiplying a fraction hy a frac- tion^ the process consists of two steps ^ either of which may be taken first ; that^ in many cases ^ there are two loays of per- forming each part of the process^ on account of the two numbers in the multiplicand^ but that, of the tivo ways, that is to be adopted which loill give the result in the loioer terms ; that each p%rt of the process is to be expressed and explained separately ; and finally, that the process is to be performed by reducing the expression of the result to its simplest terms. 1^* Exercises in multiplying Fractions by Fractions. In like manner, solve and explain the following problems. 1. If a benevolent man, having only ^ of a bushel of wheat, should give | of it to his poor neighbors, what part of a bushel would he give away? 2. At y of a dollar a yard, what part of a dollar would -J of a yard cost ? 3. What number is equal to it ^^ nr- ^ 4. If a yard of cloth cost 5^ dollars, what would |- oi^ a yard cost ? 5. At f of a dollar a yard, what will f of a yard cost ? 6. At I of a dollar a pound, what will J of a pound of tea cost ? 7. At ^ of a dollar a pound, what will f of a pound of coffee cost ? 8. At 2 J. dollars a bushel, what will 6^ bushels of wheat cost ? 9. At /^ of a dollar per hour, how much may be earned in J of an hour ? 104 ARITHMETIC. 10. At 6| dollars a barrel, what will ^^ of a barrel of flour cost ? 11. If 7| yards of satinet be bought at | of a dollar per yard, what would be the whole cost ? 12. If 1 cord of wood cost 6| dollars, what will 7| cords cost? 13. At i^ of a dollar a pound, what will 17| pounds of sugar cost ? 14. At 3|- shillings a yard, what will 8| yards of ribbon cost ? 15. If 1 dollar buy | of a gallon of wnne, how much would 67^ dollars buy ? 16. What is the value of 36f acres of land, at 40^ dol- lars per acre ? 17. What is the value of 142J tons of coal, at 7f dollars per ton ? 18. What is the value of 16| tons of hay at 11^ dollars per ton ? 19. What will 7|| bushels of apples cost at -JJ of a dollar per bushel ? 20. A merchant owning ^^ of a ship, sold f of his share; what part of the whole ship did he sell ? 21. What is f of I- ? 22. Multiply ^^ by f 23. Multiply I of f by ^-3-. 24. What is -fV of f of I ? 25. What is ^^ of ^f multiplied by f f ? 26. Ascertain the product of the following factors, J X X I X t- 27. How much is f of | of | of | ? 28. Multiply 8| by f 29. Multiply ^\ by 173-3^. ;30. Multiply IIH by SxV 31. What is the second power (49) of |? 32. What is the second power of ? 33. What is the third power of | ? 34. What is the fourth power of | ? ISl. Illustration of the Principle of dividing by a Fraction. If a philanthropist have eight dollars to distribute to the poor, to how many persons could he give 4 dollars apiece ? FRACTIONS. 105 2 dollars apiece ? 1 dollar apiece ? ^ of a dollar apiece ? ^ of a dollar apiece ? He could give 4 dollars apiece to as many persons as there are times 4 dollars in 8 dollars. 8 -r- 4 = 2 i^ersons. He could give 2 dollars apiece 8 -=- 2 = 4 persons. to as many persons as there are 8 -T- 1 = 8 persons. times 2 dollars in 8 dollars. 8 X 2 = 16 persons. He could give 1 dollar apiece 8 X 4 = 32 persons. to as many persons as there are times 1 dollar in 8 dollars. He could give J of a dollar apiece to as many persons as there are times | of a dollar in 8 dollars ; and, since there are 2 halves in every unit, (S4L^) there will be 2 times as many halves as units ; therefore, multiply 8 by 2 to ascertain how many times J is contained in it. He could give | of a dollar apiece to as many persons as there are times J of a dollar in 8 dollars; and, since there aie f in a unit, there will be 4 tiqies as many Js as units; therefore, multiply 8 by 4 to ascertain how many times J is cc'ntained in it. Observe, that, since the divisor shoivs how many equal parts, sixch as the quotient, will make the dividend; (63) when the divisor is 1 the quotient will not differ from the dividend ; when the divisor is greater than 1 the qvjotient will he less THAN THE DIVIDEND ; hut when the divisor is less than 1 the quotient will be greater than the dividend. 1^9* Model of a Recitation. 1. What would be the price of 1 acre of land, if 25 dollars be paid for 6 acres ? for f of an acre ? for 4f acres ? If 6 acres be bought, paying one dollar per acre would 25 - 6 = 4i dollars. f ^"''"t ^"^^^'^ ' *'''"'- ^^ = i^a = 33i dollars. f"""^', * ?"*= P^^ f"^ 3 * "* would be as many dollars as 6 dollars is contained times in 25 dollars. If I of an acre be bought, paying 1 dollar per acre would require J of a dollar; therefore, the price per acre would be as many dollars as f of a dollar is contained times in 25 dol- lars. Since there are 4 times as many Js as units, (08^) in any number, multiply by 4 to ascertain how many times j is contained; then, (since f is 3 times as much as J, con- sequently, will be contained only J as often as J,) divide that 106 ARITHMETIC. quotient by 3 to ascertain how many times f is contained, which reduced will be the answer required. If 4f =.A^ of an acre be bought, paying 1 dollar per acre would require -^ dollars ; -f^Xa = .j.| = 5^^ dollars. therefore, the price per acre would be as many dollars as -^ of a dollar is contained times in 25 dollars. Multiply 25 by 3 to ascertain how many times J is con- tained; (99) and, since ^ will be contained i\ as often, divide that quotient by 14 to ascertain how many times ^ is contained, which reduced will be the answer required. 2. How many barrels of flour could a trader buy for 48 dollars, at 6f dollars per barrel ? He could buy as many barrels ^8XgA=,^ = 7^ as 6f ==-2^ of a dollar is con- tained times in 48 dollars. Multiply by 3 to ascertain how many times J is contained in 48, and divide that quotient by 20 to ascertain how many times ^- is contained ; but, since 4, one of the factors of 20 is also a factor of 48 ; in dividing by 20, first divide by 4, and then divide that quotient by 5, the other factor of 20, (llTj) which will give -J of 1 = ^ of the dividend as required. In reducing this expression of the result, say 4 in 48, 12 times, and 3 times ^ are -^, equal to 7| barrels, which is the answer required. Another Explanation. First, divide by 20 as if it were 20 units, which gives ^^; but, as the right divisor is ^, only I of 20 units, it will be contained 3 times as often as 20 units (151 j) therefore, multiply that quotient by 3 to ascertain how many times ^^ is contained in 48, which gives ^9t^'^^ ==^=7-1- barrels, as before. 153* Observation. Observe, {132^) that, in dividi?ig by a fraction, the pro- cess consists of two steps, on account of the two numbers in the divisor, and that, either step may be taken first, provided the reasoning be suited to the process, 1I4. Exercises in dividing by a Fraction. In like manner, solve and explain the following problems, 1. To how many poor persons could 9 dollars be dis- tributed, giving them | of a dollar apiece ? FRACTIONS. 107 :}, If 28 dollars be paid for If Ions of hay, what is the price of a ton ? 3. If a drunkard drink ^^ of a quart of rum per day, how lor g would 9 quarts last him ? L If a moderate drinker drink | pint of brandy per day, ho AT long would 8 pints last him ? 5. How long would 2 barrels of flour last a family that consume f of a barrel in each week? 3. If 28 bushels be sown on 9J acres, how much is that pe ' acre ? 7, If it take of a bushel of rye to sow an acre, how rainy acres would 15 bushels sow? 3. How many bottles of beer holding /-g- of a gallon each, could be filled from a hogshead holding 6S gallons ? 9. At 1^ dollars a bushel, how much wheat could be bo ight for 20 dollars ? 10. How many acres would it take to produce 96 bushels, at the rate of 15f bushels per acre ? 11. If a man pay 21 dollars for pasturing his horse 16 weeks, how much is that per week? 12. If a man earn 6 dollars in f of a month, how much is theit for one month ? 13. In what time can a man build 28 rods of wall, if he build f j of a rod per hour? 14. If IJ yards of cloth be put into a coat, how many coats may be made from 30 yards ? 15. At I of a dollar a bushel, how many bushels of corn may be bought for 125 dollars ? ^ 16. How many pairs of gloves may be bought for 12 dol- lars at f of a dollar a pair ? 17. If 7|f barrels of apples be bought for 20 dollars, what is the cost of one barrel ? 18. If ll-j^j- gallons of molasses cost 3 dollars, what would be the cost of one gallon ? 19. Divide 128 by ^^, 20. How many times is ^*- contained in 19 ? 21. How many times is f contained in 14? 22. How many times is ^ contained in 9 ? 23. Divide 2 bv7|. 24. Whatpart*'of7is3? 25. What part of 7 is f ? 26. What part off is 7? 108 ARITHMETIC. 27. What part of 2^ is 2 ? 28. What part of 6 J is 5 ? 1S5. Model of a Recitation. 1. With 3f dollars how many yards of broadcloth, at 9 dollars per yard, could a merchant buy ? How many yards of cassimere, at 2 dollars per yard ? How many yards of satinet, at J of a dollar per yard? How many yards of camlet at |- of a dollar per yard? How many yards of velvet at 2f dollars per yard? He could buy as many yards of broadcloth, at 9 dollars yi:- = i yards of broadcloth. S n ' ^ ^ ^ -^ dollars is con- 5^2- == fi^ = Hi ya^ds of cassimere. tained times in 2 7-i^3 3^ 9 = 41 yards of satinet. ^^i^' ^^^\^^^ 9 ~* -^ 2 J which ascertain y-iisxb = ^^ = ^ yards of camlet. by dividing the P^-B = U=-Hi yards of velvet. ;-*- J ^f^ He could buy as many yards of cassimere, at 2 dollars per yard, as 2 dollars is contained times in 3f = -^- dollars ; which ascertain by dividing the size of the parts by 2, (II45) that is, making the parts J as large. He could buy as many yards of satinet, at | of a dollar per yard, as | of a dollar is contained times in ^-^- of a dollar ; multiply by 4, by making the parts 4 times as large, (IOO5) to ascertain how many times ^' is contained in ^^-; (Itil ) and, since | is 3 times as much as ^, and, con- sequently, will be contained only -J- as often as ^, divide this quotient by 3, by dividing the number of parts, to ascertain how many times | is contained, which reduced will be the answer required. He could buy as many yards of camlet, at f of a dollar per yard, as | of a dollar is contained times in 3f = %7- of a dollar ; multiply by 8, by multiplying the size of the parts, to ascertain how many times |- is contained in -^/-, and, since f will be contained -J as often, divide this quotient by 5, by dividing the size of the parts, to ascertain how many times | is contained, which reduced will be the answer required. He could buy as many yards of velvet, at 2=:| of a dollar per yard, as f of a dollar is contained times in 3f = ^Z- of a dollar ; multiply the number of parts by 3 to ascertain how many times J is contained in ^, and divide the size of FRACTIONS. 109 the parts in that quotient by 8 to ascertain how many times f is contained, which reduced will be the answer required. Another Explanation. First, divide 3f , or %7_ by 8 as if it were 8. units, which gives -g^^; but, as the right divisor is f, orly ^ of 8 units, (94) it will be contained 3 times as often as 8 units ; therefore, multiply that quotient by 3 to ascertain he w many times f is contained ; which gives %^-f = f^ = ^ii yards as before. 1I6 Observation. Observe, that, in dividfng a fraction hy a fraction^ the process consists of two steps, either ofiohich may be taken first ; thiit, in many cases there are two loays of performing each part of the process, on account of the two numbers in the divi- de. id ; but that, of the tiuo ways, that is to be adopted which wiR give the result in the loiver terms ; that, each part of the process is to be expressed and explained separately; and fit'jolly, that the process is to be performed by reducing the expression of the result to its simplest terms, 1>7 Exercises in dividing a Fraction by a Fraction. In like manner, solve an^ explain the following problems, 1. At f of a dollar a bushel, how much rye may be bought for 5 of a dollar ? 2. At 1^ of a dollar a bushel, how many apples may be bought for |- of a dollar ? 3. How many bushels of turnips, at ^^ of a dollar per bushel, may be bought for J of a dollar ? 4. If 1 bushel cost :| of a dollar, how many apples may be bought for f of a dollar ? 5. At J^ of a dollar a dozen, how many dozen of lemons may be bought for If dollars ? 6. At f of a dollar a dozen, how many oranges may be bought for 5f dollars ? 7. At ^ of . a dollar a pound, how many figs may be bought for 2J dollars ? 8. At ^ of a dollar a bushel, how many potatoes may be bought for 4^ dollars ? 9. At f of a dollar a bushel, how many onions may be bought for \ of a dollar ? 10. With 53 dollars, how many pounds of butter, at j*^ of a dollar a pound, may be bought ? w 10 AEITHMETIC. 11. If f of a pound of fur is sufficient for 1 hat, how many hats would 4^^ pounds be sufficient for ? 12. If 1 yard of linen cost f f of a dollar, how much would 3| dollars buy ? 13. If If yards of cloth make 1 coat, how many coats may be made from 9j- yards ? 14. If 2^ bushels of oats keep 1 horse a week, how many horses will 1S| bushels keep for the same time ? 15. If 2J- bushels of oats keep a horse 1 week, how long would 12| bushels keep him ? 16. Bought 3^ yards of cloth for 14|- dollars; what did I give per yard ? 17. At f of a dollar a pound, how many pounds of coffee may be bought for 12J dollars? IS. If 4f pounds of butter serve a family 1 week, how many weeks would 36|- pounds serve them ? 19. If a man walk a mile in ^ of an hour, how far would he walk in 5f hours ? 20. If a barrel of cider last a cider-drinker S^ months, how many barrels would he drink in lOf months ? 21. If the stage run 8y^ miles per hour, how long would ' '*^ be in running 25^^ miles ? 22. How many bushels of rye at f of a dollar per bushel, may be bought for 12/3- dollars ? 23. If 4j- pounds of t^a cost 32-^^ dollars, what is that per pound ? 24. How many times is 4^ contained in 3^? 25. At If dollars per yard, how much carpeting may be purchased for 33^ dollars ? 26. Divide If by 33f 27. Divide 33^ by 1^. 28. If ^f of a dollar buy a pound of tea, how much would 3^ dollars buy ? 29. How many times is 16 contained in 83^ ? 30. How many times is 6^ contained in 62^ ? 31. How many times is 8 J contained in 66f ? 32. How many times is 18| contained in 37j ? 33. How many times is 4^ contained in 33^ ? 34. At J of a dollar a bushel, how much corn can be bought for ^ of a dollar ? 35. At 3 dollars a yard, how much velvet may be bought for I of a dollar ? 36. Divide^ by 3? FEACTIONS. Ill 37. What part of 10 is 7 ? 38. What part of 3 is | ? . 39. What part of 3 is 2^ ? 40. Divide 5^ by 10. 41. What part of 10 is 5^? 42. Divide 2f by 7f . 43. What part of 7f is 2f? 44. Divide f by ^. 45. When corn is |- of a dollar per bushel, what part of a bushel may be bought for f of a dollar ? 46. f is what part of J ? 158. Review of the several Ways of multiplying A Fraction by a Fraction. Multiply -f^ by f . (a.) State the problem. bo, 1. 1^^ == -J. (b.) What may be the first step ? * 2. ^^p^ = yV = h (c.) Why need that be don6 ? 3. 352x-5=T=A==-^- W How may that be done? 4. f^^ == 20 = |. (e.) Why may it be done in that way? (/.) Result of the first step? ** ^- -fsii = i' (o-) What must be the next step? ^i 6. xV^^^ = xt = i- (^O Why must that be done ? " 7. i^r4><5 = A- = i- ( ^- ) How may that be done ? " 9- xVSf = = -i- U') Why may it be done in that way? {k.) Result of both steps? 1^0 Model of a Recitation. No. 1. (a.) The product should be ^ of the multiplicand, (144.) (b.) First divide by 5, (c.) to obtain (92) h W which is done by dividing the numerator, (111) by 5; (e.) because that will give \ as many parts, (/.) or -j^^' (S-) Next, multiply by 4, {h.) to obtain |, [i.) which is done by dividing the denominator (100) by 4 ; [j.) because that will make the parts 4 times as large, {k,) or ^, which is the answer required. In like manner explain the first four of the examples ; but explain the last four by ttsing the numerator of the multiplier in the first step of the process. 112 ARITHMETIC. 160. Review of the several Ways of Dividing a Fraction by a Fraction. Divide Jf by f . No. ^' ifcisxT == iS = f 4. X^^^ _ 60 _ -TU"(y- ( = m^ = nn^ = inU> &c. this taWe, that ! 1 unit is re- * ' "tVj = tVtt^ = TTHj^j = TyW(j' &c. duced to tenths, \ \ * ' ^c, ^ to hun- ' ' ; TiTyj = T*^TTJ = Ti8wj &c. dredths, &c., ' ' y^^ to ^AOM- [ ! I [ tttW> = TiTTjVorj <^c. sandths, &c., . . . ; yx>aTy to ifew- ;;;;'. TTTTyxru' ^^' thousandths, 1.1111. &., by mul- tiplying both terms ( 121 ) of each fraction by 10, and by 10 again, &:c. ; that 1 part of each size makes 10 parts of the next smaller size, or that 10 parts of each size make 1 part of the next larger size, (10) ; and that on the left, 1 part of each, size is arranged, without its denominator, according to the values of the parts, 1 unit being written, then 1 tenth in the first place at the right hand of units, 1 hundredth in the second place, 1 thousandth in the third place, 1 ten-thousandth in the fourth place, &c. Any other digit written in any of these places, would express parts of the size for which that place is appropriated. Hence, the values of these parts, or any num- ber of parts arranged in this way, according to their values, may be known without their denominator, since the different parts will always occupy places at the same relative distances from the unit's place ; but a point ( ) must be prefixed to a fraction to distinguish it from an integral number, or the integral part of a mixed number. Such fractions are called Decimal Fractions, because the parts expressed by them are always such, that it takes 10 of them, or some power (40) of 10 to make a unit. They differ from Common Fractions, only in the uniformity in the values of the parts expressed by them, and consequently, in the manner of writing them, and operating by them. 164. Mode of Reading Decimal Numbers. Since, as you may observe, the places equidistant from the units, on each side, correspond in name, except that the ter- DECIMAL fractions". 116 mination of the fractional names is ths^ the manner of reading decimal fractions is similar to that of reading integral numbers. Observe^ also, in the table, that 1 = 10000 ten-thousandths, j-^^-= 1000 ten-thousandths, xi-(j== ^^^ ten-thousandth, y^Vrr ==10 ten-thousandth, andY^^^=l ten-thousandth; con- sequently, the whole mixed number, 1.1111, may be read, eleven thousand one hundred and eleven ten-thousandths, piecisely the same as an integral number, except at last, speaking the denominator of the last figure, which is also the C('mmon denominator of this and the other figures in the number, as may be observed in the table. But the better way is to read the integral and fractional parts separately. Thus : One, and one thousand one hundred and eleven ten-thousandths. The denominator of the last figure, or the common denomi- nator of all the figures in the numerator, may be known from tie fact that it will always consist of one more figure than the decimal places occupied by the numerator, or 1 with as many ciphers as the numerator occupies decimal places. 165 Exercises in Reading Decimal Numbers. In like manner, read the following numbers. L 2. 3. 4. 5. 6. 7. 8. 9. 10. 5.111. 3.12. 2.6. .2. .25. .75. .125. 17.3. 144.16. 3456.4. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 252.5. 25.25. 2.525. .2525. 40.5. 4.05. .405. 306.1. 30.61. 3.061. 21. .05. 31. 2.40003. 22. .005. 32. 2.400305. 23. .0005. 33. .5. 24. .00005. 34. .50. 25. .000005 35. .500. 26. .007. 36. .5000. 27. .00007. 37. .050. 28. .072. 38. 8.0900. 29. 3.0407. 39. .0000009. 30. cim. 3.4007. iL NUMBE 40. rs. 1.00000080 106. Mode of writing Decimal In writing a decimal fraction, it should be made to occupy as many places as it requires ciphers in its denominator. Therefore, following the point, write as many ciphers as the number of decimal places required exceeds the number of figures expressing the numerator ; then* write such figures as will express the numerator; and the fraction will be properly expressed. ^ 116 A.RITHMETIC. 107. Model of a Recitation. Write twelve, and one thousand and sixteen ten-niil- lionths. First, write twelve and the point, thus ; 12. ; then, as the decimal must occupy seven places, and it requires only four figures to express the numerator, write three ciphers, and one thousand and sixteen, thus; 12.0001016, which is the number required. 168. Exercises in writing Decimal Numbers. In like rrunvner^ write the following numhen^ expressing the fractions decimally. 16. tVttV 18. Tf(T. 21. Seventeen, and four hundred and nine thousandths. 22. Six, and sixty-five thousandths. 23. Seven, and seven ten-thousandths. 24. Ten thousand eight hundred and nine hundred-thou- sandths. 25. Twenty-six, and fifteen millionths. Three, and one hundred and one ten-thousandths. Four, and twenty-five hundred-thousandths. Eight, and six hundred and four millionths. One, and sixty thousand and five ten-millionths. Two, and thirty thousand hundred- thousandths. How many thousandths in .2? How many hundredths in 2.5 ? Reduce -fjy to thousandths. Reduce xV^^V to its lowest terms. Reduce .25 to its lowest terms in a common fraction. Reduce .3125 to its lowest terms. Reduce ^, y^, and y^^j to thousandths and add 1. 27^^. 6. I^xtjVtt. 11. ISx^TT. 2. 14x5x7. 7. A- 12. It**!.. 3. 108xV. 8. T^xy. 13. 17x||^. 4. 73xTr. 9. TTrxrxr. 14. TTTTTO'TJ'. 5. ^100* 10. nj-fuu". 15. ^- 26. 27. 28. 29. 30. 31. 32. 33. 84. 35. 36. 37. them. 38. and add them. Reduce -fj^ , xttWj ^^^ x^j to a common denominator, DECIMAL FRACTIONS. 117 1>9 Federal Money expressed by Decbial Numbers. -federal money is the metallic money which is coined by the authority of the United States. It consists of eagles, dollars, dimes, cents, and mills ; the values of which, as you may observe in the folio v/ing table, correspond to decimal nu nbers ; 1 coin of either denomination equaling 10 of the next lower; or 10 coins of either denomination equaling 1 of the next higher, ( 10.) But the mill is only an imaginary coin. n commerce, eagles are expressed in dollars, and dimes in ceiits. The dollar is considered the unit, and cents and mi Is, decimal fractions of a dollar. Hence, numbers express- ing- Federal money are precisely like numbers in decimal fra:tions, and they are made to express Federal money by prefixing to them this character ($.) 2 ri . J 2 J ^ Q p o S '^^ Read, 1 = 10= 100 = 1000 = 10000 = $10. ... Ten dollars. 1 = 10 = 100 = 1000 = $ 1. . . . One dollar. 1= 10= 100 = $ .10. Ten cents. 1= 10 = $ .01. One cent. , 1 = $ .001 One mill. 11111 (fljil 111 5 Eleven dollars, eleven lllli ^11.111 ^ cents and one miU. 170* Reduction of Federal Money illustrated. 1. In 25 how many hundredths? how many thous- andths ? Since there are ^%% in 1, there 2500 hijpdredths. will be 100 times as many hun- dredths as units ; therefore, multi- 25000 thousandths. ply the units by 100, by annexing two ciphers, (3^1:.) There will be 1000 times as many thousandths as units; therefore, annex three ciphers to 25, which will give the answer required. 2. In $25, how many cents ? how many mills ? Since there are 100 cents in $1, there 2500 cents. will be 100 times as many cents as dol- lars ; therefore, annex two ciphers, (10) 25000 mills. to 25, making 2500 cents, which is the answer required. lis ARITHMETIC. There will le 1000 times as many mills as dollars; there- fore, annex three ciphers to the dollars, which will reduce the dollars to mills, as required. Observe that^ hij 'pointing off the ciphers annexed in these examples^ that is, putting a point between the 25 and the ciphers, the hundredths and thousandths will be reduced to units again, and the cents and mills to dollars again, 171. Model of a Recitation. 1. In 25125 mills how many cents ? how many dollars ^ Since in 1 cent there are 10 mills, 2512.5 cents. there will be y^, or .1 as many cents as mills ; therefore, divide by 10, by point- $25,125 ing off one figure at the right hand, ( 10) ; for thus the tens become units, and the other figures, also, are all brought one degree lower There will be tx^xj-, or .001 as many dollars as mills , therefore, point off three figures, (10) ; for thus the thousands become units, and all the other figures are also brought three degrees lower. 173. Exercises in the Reduction of Federal Money. In like manner, solve and explain the following 'problems, 1. In$16, how many cents? 2. How many mills in $16 ? 3. In 12000 mills how many cents ? 4. How many dollars in 12000 mills ? 5. In 75 cents how many mills ? 6. In $8.25 how many cents ? 7. In $5,125, how many mills ? 8. How many cents in $5,125? 9. In $3,375, how many dollars, cents and mills ? 10. In 16125 mills how many dollars? 11. In 12548 cents, how many dollars ? 12. Reduce $37.50 to cents. 13. Reduce 75625 mills to dollars ? 14. Reduce 984 mills to dollars. 15. Reduce $.75 to cents. 16. Reduce $.125 to mills. 17. Howmany mills in $1.25? 18. How many cents in $2,375? 19. In 12345 mills how many dolUrs, cents and mills ^ 20. How many times 10 in 85 ? DECIMAL FRACTIONS. 119 21. Divide 625 by 100. 22. Divide 1836 by 1000. 23. What is y^^ of 1728? 24. How many times 100 in 1276 ? 25. Reduce 12.25 to hundredths. 178. Model of a Recitation. 1. Bought 1 barrel of flour for $6.75, 10 pounds of cof- fee for $2.20, 7 pounds of sugar for $.875, 12 pounds of bjtter for $2, 1 pound of raisins for $.125, and 2 oranges for $.06. What was the whole amount ? Arrange the numbers together so that the figures of each denomination, may stand in a column by them- 6.75 selves, and proceed as in the addition of 2.20 integral numbers, (20.) .875 The 10 mills of the first column make 1 2. cent, (I6O5) which added with the first column .125 of cents make 21 cents, equal to 1 cent, which .06 write, and 2 dimes ; which add with the other dimes, making 20 dimes, equal to 2 dollars ; $12.01 write a cipher in the dimes' place, or second place of cents, and add the 2 dollars 'with the other dollars, making 12 dollars, which written at the left of the point, make $12.01, the answer required. 2. Mr. Farmer having a pasture of 25 acres, fenced off 2.375 acres to plant with potatoes ; hoV many acres remained in the pasture? Write the subtrahend under the minuend, placing the figures of each denomination under those 25. of the same denomination, and proceed as 2.375 in the subtraction of integral numbers, {33^.) Since there are no thousandths 22.625 acres. from which to take the 5, reduce 1 of the 5 units to tenths, (IO5) making 10, one of which (leaving 9,) i:educe to hundredths, making 10, one of which (leaving 9,) reduce to thousandths, making 10, from which subtract the 5, and 5 thousandths remain, which write ; 7 hundredths from 9 hundreths leave 2 hundredths, which write ; 3 tenths from 9 tenths leave 6 tenths, which write; 2 units from 4 units leave 2 units, which write; and blank from 2 tens leaves 2 tens, which t^rite; making 22.625 acres, which is the answer required. 3^ 120 ARITHMETIC. 174. Exercises in adding and subtracting Decimal Numbers. In like manner^ solve and explain the following problems y taking care to keep a point between the integral and frac- tional parts of every number. 1. Bought a pair of oxen for $76.50, a horse for $75, and a cow for $25.75; what was the whole amount? 2. A man gave $4.75 for a pair of boots, and $2.25 for a pair of shoes ; how much more did the boots cost than the shoes ? 3. A man bought a cow and calf for $28,375, and sold the calf for $3,625, what did the cow cost him ? 4. Bought a horse for $92, but sold him so as to lose $15.25 ; for how much was he sold ? 5. What is the whole cost of a cart at $17,625, a wagon at *S.50, a plough at $7,333, a rake at $.42, a hoe at $.60, and a pitchfork at $.875 ? 6. How much cloth in 6 pieces measuring as follows, 25.5 yards, 27.75 yards, 28.125 yards, 30 yards, 29.375 yards, and 26.5 yards ? 7. A merchant having a piece of cloth measuring 25 yards, sold from it 1.875 yards for a coat, and 1.125 yards for a pair of pantaloons ; how much was there left in the piece ? ' 8. Mr. Farmer took to market 32 bushels of potatoes in one load, and peddled them as follows : 5.3 bushels for $2.75, 4.25 bushels for $2,125, 6.75 bushels for $3,375, 10.5^ bushels for $5.25, and the rest of the load for $2.50; how ^ much did he sell at the last sale ; and how much did he get for his load ? 9. A man owing $253, paid $187,375, how much did he then owe ? 10. Add together 10.0625, 5.1875, %.5, and 4.25. 11. How much is 15.5 + 2.75 + 3.75 12 ? 12. What is the sum of 192.423 and 20.58? 13. What is the difference between 12.5 and 6.25 ? 14. ^\hat is the sum and difference of 245.0075 and 234.9925 ? 15. Subtract 2yV ^^om 4yVu 17^. Model of a Recitation. I. If 1.75 yards be required for 1 coat, how much would be required for 7 coats ? DECIMAL FRACTIONS. 121 1 175 Arrange the multiplicand and multi- ^ plier, and proceed as in the multiplication of integral numbers, (S7) and the product 12 25 vards would be 1225; but, since 7 times 175 ^ * things of any kind will be 1225 things of the same kind, 7 times 175 hundredths will be 1225 hun- dredths, or 12.25. But, to analyze it, say : 7 times 5 hundredths are 35 hun- dradths, equal to 5 hundredths, which write in the place of hindredths, and 3 tenths, (lOj) which add with 7 times 7 te iths, making 52 tenths, equal to 2 tenths, which write in th3 place of tenths, and 5 units, which add with 7 times 1 ur.it, making 12 units, which write at the left of the point, ard the result will be the answer required. 2. At $175 per acre, what would be the cost of .7 acres, or, more properly, .7 of an acre ? Since 1 acre costs $175, .7 of an acre would ?'175 ^^^^ *''' ^^^^^ ^^ much, or .7 as much. First, ry multiply by 7, as if it were 7 units, which gives $1225. But, the right multiplier being .7, only (t.-j 22 5 ro of ^ units, the right product should be only -^ of 1225 ; therefore, divide this product by 10, by removing the point one place farther to the left, (IO5) which gives $122.50, the answer required. 3. What would be the price of 2.25 cords of wood, at $5,375 per cord ? Since 1 cord costs $5,375, 2.25 cords $5,375 would cost 2.25 times as much. 225 2.25 times 5.375 would be 1209.375. But, ~26875 ^^^ multiplier being only y^^ of 225, the 107^50 product will be only y^ of 1209.375; 10750 therefore, divide by 100, by pointing off two more figures (171) for decimals, $12.09375 making $12.09375, which is the answer required. 4. Multiply .125 by .03. 3 times .125 would be .375. But the multi- .125 plier being ^-^^ of 3, the product will be ^-^^ .03 of .375 ; therefore, divide by 100, by removing the point (163) two place farther to the left. .00375 But you must make those places in this exam- ple, by prefixing ciphers. 11 4 122 arithmetic. 170* Proof of the Pointing in the Multiplication of Decimals. Observe, ihat^ in the preceding examples, {V75^) each product has as many decimal figures as all its fg,ctors. This will hold true in all cases ; and this tfuth may he applied to prove the pointing of the product ; for, if the factors be con- sidered as integral numbers, the product would be integral ; and, since for every removal of the point one place to the left, in either factor, that factor becomes y\j- as large, (lO,) and, consequently, the product also becomes yV ^^ large, the pro- duct must be divided by 10, [which is done by removing the point (10) on place to the left,) for every decimal figure IN ALL THE FACTORS. lyy. Exercises in the Multiplication of Decimal Num- bers. In like manner, solve and explain the following problems, 1. How many yards of cloth would be refjuired for 5 pairs of pantaloons, if 1.25 yards be put into each pair ? 2. What cost 8 yards of cloth, at S2.875 per yard ? 3. How many dollars in 8 ninepences, if $.125 make 1 ninepence ? 4. If $.0625 make 1 fourpence-halfpenny, how much ia 16 fourpence-halfpennies ? 5. How much would a man receive for 5 barrels of pork at $17.25 per barrel ? 6. At $5.50 per yard, what cost 10 yards of broad cloth ? 7. At $.05 per pound, what cost 100 pounds of rice ? 8. At $.20 per pound, what cost 1000 pounds of butter ? 9. What cost 60 pounds of candles, at $.17 per pound ? 10. What cost 12 dozen of eggs, at $.125 per dozen ? 11. Multiply 5.333 by 8. 12. Multiply .464 by 25. 13. How much is 50 times .05 ? 14. What is the amount of the following bill ? Mr. John Debtor, 1^^^"' "^""^^ ^' ^^^- Bought of Charles Creditor, 7 yds. Broad Cloth, $5.50 per yard, 5 " Cassimere, (a> 1.50 " " 12 *' Striped Jean, (cb .375 " " 15 " Bleached Sheeting, ( A^ " '' 27 '' Brown " (t .125 '' " DECIMAL FRACTIONS. 123 15. If a barrel of flour cost $6, what cost .5 barrels ? 16 At $25 a ton, what cost .7 of a ton of hay ? 17. At $6 per yard, what cost .25 of a yard of cloth ? 18. At $8 per cord, what cost .75 of a cord of wood ? 19. At $5D per acre, what cost .125 of an acre of land ? 20. What is the amount of the following hM ? Mr. Jacob Shem, ^^l^-"' ^""^ ^' ^^^^- Bought of Israel Ham, JJ7.5 yards German Broad Cloth, < $9 per yd. tl5.75 " French " " o $7 " " 18.125 " English Cassimere, $3 " " IM.375 " American " $2 " " 21. Multiply 144 by .5. 22. What is .5 of 1728? 23. Multiply 512 by .25. 24. What is .75 of 856? 25. Multiply 1840 by .125. 26. What is .625 of 1000 ? 27. Multiply 75 by .004. 28. What is .0003 of 3000? 29. At $.96 a gallon, what costs .4 gallons of oil ? 30. At $.50 a yard, what costs .5 yards of cloth ? 31. Multiply .3 by .6. 32. What is .4 of .7 ? 33. Multiply .25 by .5. 34. What cost 1.5 yards, at $.12 per yard ? 35. What cost 4.12 yards, at $.50 per yard ? 36. What is the amount of the following bill ? Mr. Reuben Retail, ^^'"' ^""^ ^' ^^^' Bought of Warren Wholesale, 6.5 dozen Spelling-Books, ( $ 1.75 per doz. 8.25 " Young-Readers, ( $ 2.875 " " 10.75 " National-Readers, (8> $ 7.50 " " 9.25 " Testaments, (Q $ 3.625 " " 4.5 " Polyglot Bibles, ($10.50 " 124 ARITHMETIC. 37. What is the product of .204 multiplied by 1.4 ? 38. How much is 11.03 times .1109 ? 39. Multiply .04 by .004. 40. What is .0006 of .0012 ? 41. Multiply 1.006 by .002. 42. Multipl||> .062 by .003. 43. How much is .0004 of .025 ? 44. What is the second power (49) of .5 ? 45. What is the second power of .25 ? 46. What is the third power of .5 ? 47. What is the fourth power of .5 ? 178. Model of a Recitation. 1. If 4 books cost $3, how much would that be apiece ? One book being J of 4 books, the price 4) 3 rS 75 ^^ ^ ^^^^ should be J of the price of 4 2 8 * books. ^ of 3 dollars would be | of a dol- lar, (94:5 ) ^^ ^ common fraction ; but the 2Q answer may be obtained in a decimal 2Q form. Thus, J of 3 dollars not being a whole dollar, reduce the 3 dollars to dimes, or tenths, (ITOj) making 30 tenths, J of which is .7, and 2 dimes, or tenths, re- maining, which reduce to cents, or hundredths, making 20 hundredths, J of which is .05, which, written with the .7, makes $.75, the answer required. 2. A man, having 3 acres of land, divided it into 8 equal house-lots. How many acres in each lot ? g Each lot would contain J of 3 Q\ Q n / Q-r^ ,.^o acres, which is of an acre ; but o) J.U (.J7o acres. ^i- r . -l j j OA this common traction may be reduced to a decimal fraction. Thus, J of 3 QQ acres not being a whole acre, reduce en the 3 acres to tenths, making (163) 30 tenths, J of which is .3, and .6 ^Q remaining, which reduce to hun- Ar. dredths, making 60 hundredths, ^ of which is .07, which write, and .04 remaining, which reduce to thou- sandths, making 40 thousandths, J of which is .005, which write, making .375 of an acre, the answer required. DECIMAL FRACTIONS. 125 3. At $3,375 for 9 gallons of molasses, what would be the cost of 1 gallon ? One gallon would cost -J of the price of 9 (,v o rt^c gallons. ^ of 3 dollars not being a whole dol- *^ ' _J lar, reduce the 3 to tenth s^ making, with the 3 <^ rt^^ tenths, 33 tenths, ^ of which is .3, and .6 re- maining, which reduce to hundredths, making, with the 7 hundredths, 67 hundredths, \ of wiich is .07, which write, and .04 remaining, which reduce to thousandths, making, with the 5 thousandths, 45 thou- sandths, \ of which is .005, which write, making $.375, which is the answer required. ITO. Exercises in reducing Common Fractions to Decimal Fractions. In like manner^ solve and explain the following problems. 1. If 12.25 yards be required for 7 coats, how much W3uld be required for 1 coat ? 2. If $.375 be paid for 3 boy's tickets for admission to a concert, what would be the price of 1 ticket ? 3. If 4 rides in a car cost $6, what is the cost of 1 ride ? 4. If 3 bushels of apples be divided among 4 men, what would be each man's share ? 5. If 12 dozen of eggs cost $1.50, how much is that per dozen ? 6. If 3 acres of land be fenced off into 5 equal parts, how many acres in each part ? 7. Reduce f to a decimal fraction. 8. Reduce | to a decimal fraction. 9. Reduce | to a decimal fraction. 10. Reduce ^ to a decimal fraction. 11. Reduce -f^ to a decimal fraction. 12. How many times is 24 contained in 6 ? 13. How many times is 12 contained in 28.8 ? 14. Divide 17.28 by 48. 15. Divide 1.44 by 72. 16. Divide 4.096 by 64. 17. Reduce ^VV ^^ ^ decimal fraction. 18. How many times is 50 contained in 2.5 ? 19. How many times is 100 contained in 5 ? 20. Divide 3.75 by 8. 21. Divide 2.5 by 4. 11# 126 ARITHMETIC. 180. Model of a Eecitation. 1. Divide 54.32 by 40. The factors 10 and 4 composing 40, first A) 5 4*^2 divide by 10, by removing the point (10) one _^ place farther to the left, to obtain -^jj of the 1 rjro dividend, which divide (118) by 4, to obtain 4 of tVj 0^ A- of the dividend, which will be the answer required. 2. How many times is 35000 contained in 31.5 ? The factors 1000, 7, and 5, composing 7^ A01C 35000, first divide by 1000, by removing the ^ '_ point (10) three places farther towards the 5^ 004-^ ^^^' ^^ obtain iwau of the dividend, which ^ '_ '_ divide by 7, to obtain ^ of xxrW' or ytjVd /^Q/^Q (118) of the dividend, which divide by 5, to obtain -^ of tuVtu or -g^^iyxr of the dividend, which will be the answer required. 181. Observation. Observe that, when the divisor is a certain number of tens, hundreds, or thousands, ^c, it is more convenieiit to divide first by one ten, hundred, or thousand, <^c.; then divide that quotient by the other factor of the divisor. 182. Exercises in dividing by Units of the higher Orders. In like maimer, solve and explain the following problems, 1. How many times is 500 contained in 1775 \ 2. Divide 6.25 by 250. 3. How many times 9000 in 63459 ? 4. What is the quotient of 129.6 divided by 1200 ? 5. Divide 3.651 by 30. 6. How many tons, of 2000 pounds each, in 16948.25 pounds ? 7. What part of an hour is 21 minutes ? 8. How many years would it take a man to save $5750, at $500 per year ? 9. How many months, at $60 per month, would it take a man to earn $1296 ? 10. Divide 45.6855 by 1500. DECIMAL FRACTIONS. 127 11. Divide 8943.75 by 75000. 12. What part of a ton, or 2000 pounds, are 1728 pounds ? 13. How many times is 42000 contained in 586.488 ? 14. Reduce 14784 minutes to hours. 15. How many eagles, of $10 each, in $362.50 I 1 3. Illustration of Infinite Decimals. 1. If $1 be paid for 9 writing-books, how much would that be apiece ? If 9 books cost $1, one book j would cost ^ of a dollar. But to re- 9^10C^111-U duce^ to a decimal form, (ITS) annex Q a point and a cipher to the numera- tor, and divide it by the denominator, 1Q which will give .1 for the first quo- Q tient figure. A cipher annexed to the remainder, gives .01 for the quo- 1^ tient ; a cipher anilexed to this i*- q mainder, gives .001 for the quotient; and thus, continuing without limit, 1 the same remainder would recur, and the same figure would be repeated in the quotient. This quotient, and the like, are called Infinite Bechnals. When a quotient figure thus repeats, it is called a repe- tend; and the fact of its being a repetend, is denoted by placing a point over the first figure, omitting the rest. Thus, .i = .lll&c.==|; .2 = .222 &c. = I ; .5 = .55 &c. = |; and any figure, thus repeating, is so many times -J. There- fore, to reduce any repetend of one repeating figure to a com- mon fraction, you need only make the repeating figure the numerator, and 9 the denominator, and the result will be a fraction of 1 in the next higher place. 2. Reduce -^^ to a decimal. _1^ . . When two or more figures 99)1.00(.0101 &c. = .01 repeat, as in this example, the 99 repetend is denoted by a point over both the first and last of 100 the repeating figures. Since .61 = ^^, any two figures thus repeating, equal so many thnes ^^j ^^^ i"^ ^i^^^ 128 ARITHMETIC. manner, three repeating figures equal so many times ^^, &c. Therefore, to reduce any repetend to a common fractimi^ take the repeating figures for a numerator^ and as many 95, for a denominator, and the result loill be an equal fraction of 1 in the next higher place, 184. Model of a Recitation. 1. Reduce ^ to a decimal fraction, and back again to a common fraction. i = .16 = 3V + fofTV=iV + A = /Tr + 7V = i* = i- 2. Reduce -^^ to a decimal, and back again. ^ sfA===.2Mf=A + if of3-V==//^ + ^f^===37^. 185. Exercises in the Reduction of Infinite Decimals. In like manner^ solve and explain the following problems, 1. Reduce -j^ to a decimal, and back again to a common fraction. 2. Reduce ^ to a decimal, and back again to a common fraction. 3. Reduce -{^ to a decimal, and back again to a common fraction. 4. Reduce |^ to a decimal, and back again to a common fraction. 5. Reduce f to a decimal, and back again to a common fraction. 6. Reduce Y^^xy to a decimal, and back again to a common fraction. 7. Reduce .53 to a common fraction. 8. Reduce .46 to a common fraction. 9. Reduce .325 to a common fraction. 10. Reduce .24 to a common fraction. 11. Reduce y^y to a decimal, and back again to a common fraction. 12. Reduce |-g^ to a decimal, and back again to a common fraction. DECIMAL FRACTIONS. 129 186. Model of a Recitation. ' . Reduce ^^ to a decimal fraction. 1 It will generally be sufficiently ac- 72)1.00(.014 curate to extend the quotient only to 72 four or five places of decimals, and write in the last place the figure that 280 will make the quotient the nearer cor- 288 rect, with ( ) after it if the figure be too large, and (-f-) if it be too small. But if it be required to multiply such a quDtient, some allowance should be made for its incorrect- ness. 2. How much would 125 cords of wood come to, at $5.16 per cord? /^ ,A 5 times 6 are 30, but if another 6 of the 1 Q r repetend were written and multiplied by 5, it would aflford 3 to be added to this 30, making 33; therefore, write 3 in the first place. 2 2583 times 6 are 12, but if another 6 were written 10333 ^^^ multiplied by 2, it would aflford 1 more to 51666 ^^ added to this 12, making 13 ; write the 3, and, since this 3 is a repetend, continue it to cbnA/r Qo the lowest place ; so, also, continue the 6 down to the lowest place. The sum of the first column is 12, but as there might be a lower column like this, which would aflford 1 more for this column, write 3 in the first place, &c. 3. What would be the cost of 1 pair of boots, if 5 pairs cost $16 ? 5 is contained 3 times in 16, and 1 5) 16. ($3,333 -f- remains, which reduce to tenths, making 15 with the .6 belonging to the repetend, ' 16 tenths, which will give .3 for the 16 quotient; 1 tenth remains, which re- 15 duced to hundredths, and added to the .06, will give .03 for the quotient, &c. 187. Exercises in the Use of Infinite Decimals. In like manner J solve and explain the following problems. 1 . If there are $4 in 1 sterling pound, what is the value of 1000 sterling pounds ? 130 ARITHMETIC. 2. If $.16 make a shilling, what is the value of 6 shillings ? 3. If $.083 make a sixpence, what is the value of 12 six- pences ? 4. If $.0416 make a threepence, what is the value of 24 threepences? 5. What is the .value of 100 yards of silk, at $.83 per yard? 6. If 5 spelling hooks cost $.83, what is the price of 1 of vhem ? 7. What is the value of 1 sterling pound, if 12 pounds make $53. ? 8. If 6 yards of silk cost $5, what is that per yard? 9. If 24 oranges cost $1, how much is that apiece? 10. What would 1 comh cost, if 60 combs should cost $5.00 ? 188. Model of a Recitation. 1. At $.12 a pound for raisins, how many pounds may be bought for $2.88? As many pounds may be bought .12)2.88(24 pounds. as $.12 is contained times in $2.88 ; 24 that is, 12 cents in 288 cents, or, 12 hundredths in 288 hundredths, which 48 will be as many times as 12 things of 48 any kind is contained in 288 things of the same kind; that is, 24 times ; therefore, 24 pounds is the answer required. 2. If a charitable person distribute 3 barrels of flour to the poor, giving them .125 of a barrel apiece, to how many persons could he give a portion ? To as many persons as .125 is .125)3.000(24 persons. contained times in 3. 250 Observe, that, in the first ex- ample) both divisor and dividend 500 being of the same denomination, 500, the quotient was an integral number, and necessarily so, from the fact, that if both divisor and dividend be of the same denomination, it cannot affect the quotient i whether that denomination be pounds, barrels^ miles. DECIMAL FRACTIONS. 131 ddtfSy dollars, cents, mills, tenths, hundredths, or thousandths. Fo r instance ; 6 things of any denomination are contained in 12 things of the same denomination, 2 whole times. Therefore, if this dividend be reduced to the same denomi- naion as the divisor, that is, to thousandths, the quotient so far must be an integral number. The quotient being 24, so many persons could receive a portion of the flour. 3. How many bushels of apples at $.5 per bushel may be boightfor $.375? As many bushels as .5 is contain- .).375{.75 bushels. ed times in .375. 5 tenths is not 35 contained in 3 tenths ; therefore, there can be no units in the quotient ; write 25 the point, and take into consideration 25 one more figure of the dividend, and the quotient figure thence obtained will be tenths, since tenths follow next to units ; then come hundredths, &c., in their own order. 4. At $6.25 per barrel for flour, how many barrels, or wliat part of a barrel may be bought for $.03125 ? As many barrels as 6.25 is 6. 25).03125(.005 barrels. contained times in .03125. 3125 The divisor being hundredths ' only the hundredths of the ^ dividend will aflford units for the quotient, but 625 (hundredths) not being contained in the 3 (hundredths,) there will be no units in the quotient. Write the point, and take into consideration one more figure of the dividend, which gives tenths for the quotient; the next figure gives hundredths for the quotient, and the next figure gives 5 thousandths, which, as there is no remainder, is the answer required. 189. Observation. In the division of decimals by integral, or decimal num- bers, you need have little difficulty in ascertaining the right place for the point, if you observe under staiidingly, that the figures of the dividend, as low as the lowest figure of the divisor, and no farther, loill give integral quotient figures ;, and if you are careful to write the point in the quotient as soon as you have come to its place. Should not the dividend already be as low as the divisor, make it so by annexing ciphers. 132 arithmetic. 190 Proof of the Pointing in the Division of Deci- mals. To prove whether you have pointed the quotient correctly, consider that the divisor and quotient being the two factors of the dividend, (75) must together have the same number . of decimal figures as the dividend, (1T6). If your work stands this test, probably you have put the point in its right place. 191. Exercises in the Division qf Decimals. In like manner, solve and explain the following problems. 1. How many umbrellas, at $1.25 apiece, may be bought for $3.75 ? 2. How many pairs of half-hose, at $.35 a pair, may be bought for $1.40 ? 8. How manv pounds of coifee, at $.12 a pound, may be bought for $13.44 ? 4. How many pounds of cheese, at $.07 a pound, may be bought for $5.25 ^ 5. At $1.50 per yard, how many yards of cassimere may be bought for $24. ? 6. At $.80 per yard, how many yards of kersey may be bought for $20 ? 7. At $.40 per yard, how many yards of flannel may be bought for $12? 8. At $.20 per yard, hov/ many yards of calico may be bought for $32 ? 9. If $108.50 be paid for cassimere, at $4 per yard, how many yards were bought ? 10. If $18.50 be paid for broad-cloth, at $5 a yard, how many yards were bought? 11. If $28.35 be paid for 4.5 barrels of flour, how much is that a barrel ? 12. If $153,525 be paid for 26.7 cords of wood, what would 1 cord cost ? 13. What would 1 bushel of wheat cost, if 14.75 bushels cost $18.4375 ? 14. What would lib. of sugar cost, if 375.6 pounds cost $46.95? 15. What would 1 ton of potash cost, if 28.75 tons cost $3616.175? COMPOUND NUMBERS. 133 16. What would 1 bushel of corn cost, if 63.5 bushels cc St $49.53? 17. What costs 1 yard of cloth, if 79.4 yards cost $187,384 ? 18. How much sugar, at $.125 per pound, can be bought for $15.50? 19. If 112 pounds of iron cost $7.28, what is the cost of 1 pcund? 20. How many times is $.06 contained in $33.60 ? 21. How many times is .46 contained in 18.4? 22. How many times is .18 contained in 7.02? 23. How many times is .4 contained in 2.5 ? 24. Divide 13.2 by 17.6. 25. Divide 61.512 by 2.4. 26. Divide .063 by 10.5. 27. Divide 1.8144 by 10.5. 28. Divide 5.38575 by 1.075. 29. What part of 8 is 3? 30. What part of 8 is .3? 31. What part of 2.4 is .6? 32. What part of 10.35 is 5.175? 33. How many times is .2 contained in .06 ? 34. How many times is .04 contained in .008? 35. Divide .00003 by .003. 36. Divide .000011021 by .0107. 37. Divide .00001 by .025. 38. Divide 47 by .1. 39. Divide 3. by .0003. 40. What part of 1.006 is .002012? VIII. COMPOUND NUMBERS- 19^. Compound Numbers defined and illustrated. In simple numbers, such as we have heretofore employed, the several orders of units increase, or decrease, by the uniform ratio 10 or yV, that is, one unit of each order equals 10 units of the next lower order (IO5) or yV of a unit of the next higher order. But a compound number is a number which expresses a quantity in several denominatiozis having no uni- form ratio. Thus, 4 yards 2 feet 6 inches, is a compound number ; 12 inches making a foot, and 3 feet a yard. 12 134 ARITHMETIC. The ratios (26^) of the denominations of compound num" hers, are exhibited in the folhioing tables^ which should be thoroughly committed to memory. 193. Long Measure illustrated. Long measure is used in measuring distances between two points. Its unit is the mile, which is divided and subdivided according to the following Ta])le. m. fur. rods. yds. ft. in. 1 = 8 = 320 = 1760 = 5280 == 63360. 1 = 40 = 220 = 660 = 7920. 1= 51= 161= 198. 1 = 3 = 36. 1 = 12. 40 rods == 1 furlong. 8 fur. = 1 mile. 12 inches = 1 foot. 3 ft. =1 yard. 5\ yd. = 1 rod. 194. Cloth Measure illustrated. Cloth measure is used in measuring chths^ laces, ribbons, ^c. Its unit is the yard of long measure, which is divided, and subdivided, according to the following Table. yd. qr. na. in. 1 = 4 = 16 = 36. 21 inches = 1 nail. 1= 4= 9. 4 na. =1 quarter. 1= 21. 4 qr. =1 yard. 19S Square Measure illustrated. Square measure is used in measuring surfaces. The unit of measure is a square, which is a plain surface having four equal sides and angles (439). It is called a square inch, foot, yard, &;c., according as its side is one inch, foot, yard, &c., in length. If lines one inch, or foot, See, apart, be drawn parallel (4S9) to two opposite sides of a square, and, in like manner, lines be drawn parallel to the other two sides, the number of squares thus formed, or the superficial contents of the square, will be the second power (49) of the inches, or feet, fee, in a side of the square ; for the number of inches, or feet, &c., in a side of the square, will be equal to the number of rows of squares, and to the number of squares in each row, the COMPOUND NUMBERS. 136 product of which will express the contents of the square. Taus, a square foot would make 12 rows of 12 square inches each, equal to 144 square inches, which is the second pc'Wer (363) of 12, the number of inches in the side of a square foot. Hence, the contents of any rectangular surface (439) is the product of its length and breadth. The square m%le is divided and subdivided according to the following Table, n. acres, roods. rods. yds. ft. in. 1=640=2560=102400=3097600 =27878400 =4014489600. 1= 4= 160= 4840 = 43560 = 6272640. 1= 40= 1210 = 10890 = 1568160. 1= 30i= 272i= 39204. 144 inches = 1 foot. 9 ft. =1 yard. 304 yd. = 1 rod. 1 = 9 = 1296. 1 = 144. 40 rods = 1 rood. 4 roods = 1 acre. 640 acres = 1 mile. 196. Cubic Measure illustrated. Cubic measure is used in measuring solids and capacities^ 01 anything that has three dimensions, length, breadth, and thickness. The unit of measure is a cube, which is a solid having six equal square faces. (439). It is called a cubic inch, or foot, &c., according as a side of a face of it is one inch, or foot, &c., in length. If 12 boards, each a foot square and one inch thick, be piled together, they would make a cubic foot; but each board maybe divided into 12 equal pieces one inch wide and thick, and each piece into 12 cubic inches ; therefore, each board would make 12 X 12 = 144 cubic inches, and the 12 boards, or cubic fo Dt, would make 12 X 12 X 12 == 1728 cubic inches. Hence, the conte?its of a cube is the product of its three dimensions, or the product of its base (439) and height, or the third power (49) of a side of onje of the cube's faces. Hence also, the contents of any solid, SfC, having rectan^ gular faces, (439) is the product of its three dimensions, A cubic yard is divided and subdivided, according to the following table. ,. '^ 136 ARITHMETIC. Table. yd. ft. 1 = 27 = 46656. 1= 1728. 1728 inches =. 1 foot. 27 ft. = 1 yard. 128 feet make 1 cord ; but it is usual to consider |- of a cord, or 16 cubic feet, 1 cord-foot ; hence, 8 cord-feet make 1 cord. 50 feet of timber, make 1 ton. But a ton of round timber will make only 40 feet of square timber, as ^ is allowed for waste in squaring. 107* Dry Measure illustrated. Dry measure is used in measuring grain, fruit, salt, and similar dry goods. Its unit is the bushel, which is divided, and subdivided, according to the following Table. " bu. pk. gal. qt. pt. 1 = 4 = 8 = 32 = 64. 1 = 2= 8 = 16. 1= 4= 8. 1=2. One gallon contains 268|- cubic inches (190.) 198 Liquid Measure illustrated. Liquid measure is used in measuring all kinds of liquids. Its unit is the gallon, which is divided, and subdivided according to the following Table. gial. qt. pt. gill. 1 = 4 = 8 = 32. 4 gills = 1 pint. 1 = 2 =: 8. 2 pt. =1 quart. 1 =4. 4 qt. =: 1 gallon. One gallon contains 231 cubic inches. One gallon of milk, and malt liquors contains 282 cubic inches. 199. Troy Weight illustrated. Troy weight is used in weighing precious metals, and liquids. Its unit is the pound, which is divided, and sub- divided, according to the following 2 pints = 1 quart. 4 qts. = 1 gallon. 2 gals. = 1 peck. 4 pks. =: 1 bushel. COMPOUND NUMBERS. 137 TabU. oz. dwt. gr. 12 = 240 = 5760. 1= 20= 480. 1 = 24. 24 grains = 1 pennyweight. 20 dwt. = 1 ounce. 12 oz. = 1 pound. SDO* Apothecaries' Weight illustrated. Apothecaries^ weight is used in compounding medicines, lis unit is the pounds which is divided, and subdivided, accord- irg to the following TahUi 1 = 12 = 96 = 288 = 5760. 1= 8= 24= 480.' 1= 3= 60. 1= 20. 20 grains = 1 scruple. 3 9 =1 dram. 8 3 =1 ounce. 12 =: 1 pound. 901. Avoirdupois Weight illustrated. Avoirdupois weight is used in weighing coarse goods^ such as are not weighed by the Troy, or Apothecaries* weight. Its unit is the ton^ which is divided, and subdivided, according to the following Table, ton lb. oz. dr. gr. 1 = 2000 = 32000 = 512000 == 14000000. 1= 16= 256= 7000. 1== 16= 437i 1= 27H. 27H grs. = 1 dram. 16 dr. = 1 ounce 16 oz. ,= 1 pound. 2000 lb. = 1 ton. The hundred-weight and quarter are generally dispensed with ; and 2240 pounds are no longer considered a ton. The grain in the three weights is the same, but the other denominations, though agreeing in name, differ in weight, excepting in Troy and Apothecaries' weight, where they are identical. The weight of anything together with the container, is called gross weight ; and the remainder, after deduction has been made for the container, &c., is called vt weight, (307.) 12=^ . 4 # vj8 arithmetic. 202. Time illustrated. Ti7?ie is divided into years by the revolutions of the earth about the sun, and years into days by the revolutions of the earth upon its axis. A year is divided, and subdivided, ac- cording to the following Table, Year days, hours, minutes, seconds. 1 = 3654 = 8766 = 525960 = 31557600 1 == 24 = 1440 = 86400 1 = 60 = 3600 1 = 60 60 seconds == 1 minute, 60 m. =1 hour, 24 h. = 1 day, 365 J d. = 1 year. 365 J days, though not exactly a year, is sufficiently ac- curate for ordinary purposes, and will be considered a year, unless it be otherwise specified. It is usual in calendars to reckon 365 days to all years, except those divisible by 4, to which 366 days are allowed ; but centennial years, though divisible by 4, have only 365 days, except the years which are divisible by 400, which have 366 days. Years having 366 days are called leap years, in which February has 29 days, otherwise, only 28. The calendar months, April, June, September, and November, have each 30 days ; and January, March, May, July, August, October, and December, have each 31 days. But in calculations involving dates, 30 days are considered a month, and 12 months a year. 303 Circular Measure illustrated. Circular Measure is used in measuring circles, (-ASOj) and their circumferences, particularly angles, latitude and longi- tude, and the relative situations of the heavenly bodies. If from the centre of a circle^ straight lines be drawn, divid- ing the circumference into 360 equal parts, or arcs, each of these arcs is called a degree, as is also each of the spaces comprehended by two of the straight lines, or radii. Hence, a degree being -^^ of a circle, or of its circum- ference, its extent will be greater, or less, according to the size of the circle. H COMPOUND NUMBERS. 139 The divisions, and subdivisions, of a circle and its circwmr firence are exhibited in the following I Table. C. signs, degrees, minutes, seconds. 1 = 12 = 360 = 21600 = 1296000 1 == 30 = 1800 = 108000 1 == 60 = 3600 1 = 60 <)0" (seconds) = 1 minute, 00' =1 degree, 30 = 1 sign, 12s. = 1 circle. S4 English Money illustrated. English Money is the national currency of Great Britain. It was the currency of the United States till the establishment of Federal Mony, in 1786, and is partially used here at pres- ent. Its unit is the pound, which is difided, and subdivided, according to the following Table. . s. d. qr. 1 = 20 = 240 = 960 1 = 12 = 1 = 48 4 4 farthings = 1 penny. 12 pence = 1 shilling.. 20 shillings = 1. pound. SCItl. Currencies of English Money illustrated. The term pound represents different values in the differ- ent currencies ; so also do the other denominations of English money, according to the following Table, 1, Sterling, =S4|, used in England. 1 , Can. Cur, = $4, *' " Canada and Nova Scotia. 1, N. E. " =S3i, " " N.E.,Va.,Ky.,andTenn. 1, N. Y. " = $2J, " " N. Y., Ohio, and N. C. 1, Penn. "=$2|, '' " Penn.,N.J.,Del.andMd. 1, Georgia " =i4f, '* " Ga. and S.C. $1 = 4^. 6^. = ^V, Sterling. $1==55. =|j Can. Currency. $1 = 6^. =.3, N.E. " $1 = 8^. =.4, N.Y. " $1 = 7^.6^. =f, Penn. " $1 = 4^.8^. =^,Ga. M.84 is the present legal value of the pound sterling. 140 ARITHMETIC. SS06. New England Currency illustrated. Neio England Currency is still much used in appraising articles of merchandise. The most common prices are ex- hibited, reduced to Federal money, and aliquot parts of a dollar, in the following Table. S. d. S. d. 3 = 4i= S.04^ = .06J = 3 3 = 3 6 = $ .54i = .581 = ta ^ 6 = .081 = tV 3 9 = .62^ = 1 9 = .121 = i 4 = .66| = 1 1 = .16| i 4 3 = .70i = H 1 3 = 20i = ix 4 6 = .75 = J 1 6 = .25 = i 4 9 = .79i = if 19 = 2 = 2 3 = .29i = .331 = 37i = 5 = 5 3 = 5 6 = .831 = 87* = .91* = 1 2 6 = 2 9 = 3 = .41f = 45i = .50 = A 5 9 = 6 = .95f = 1.00 = 307. Model of a Recitation. 1. How many inches long is a road, which measures 3 miles, 6 furlongs, 25 rods, 2 yards and 1 foot ? Since there are 8 fur- 3m. 6f. 25rds. 2yds. 1ft. 8 30 furlongs. 40 1225 rods. 11 longs in a mile, there will be 8 times as many fur- longs as miles ; 8 times 3 are 24, which, together with the 6, make 30 fur- longs. Since there are 40 rods in a furlong, there will be 40 times as many rods as furlongs ; 40 times 30 are 1200, which, together with the 25, make 1225 rods. Since there are 5|, or -y-, yards in a rod, there will be -y- as many yards as rods ; ^ of 1225 are 6737 J, which, together with the 2, make 6739 J yards. Since 2) 13475 half-yards. 6739^ yards. 20219i feet. 12' 242634 inches. COMPOUND NUMBERS. 141 th'jre are 3 feet in a yard, there will be 3 times as many feet as yards ; 3 times 6739^ are 20218J, which, together with th3 1, make 20219| feet. And, since there are 12 inches in a bot, there will be 12 times as many inches as feet ; 12 times 2C2l9i are 242634 inches, which is the answer required. 2. Kediice 242634 inches to higher denominations. Since it takes 12) 242634 inches. J^/''?^' ^'n ' loot, there will 3) 20219 ft. 6 inches. "f f^ ^? ^^^^ ' leet as mcnes ; ri iiN^QQ A oA 1^ of 242634 ^ = V) 6739yds. 2ft. Se 20219 feet, and 6 inches re- ll)1347ahalf.yards. T^ It:. 40) 1225 rods, 3 half-yards. ^^ ^f ,^there , 8) 30 fur. 25 rods. ^7^720211 3,n.6fur.25rds.2yds.lft. T/PLTi maining. Since it takes 5|, or -^^, yards for 1 rod, there will be as many rods as -y- is contained times in 6739 ; multiply by 2, to ascertain how many times \ is contained, (IS^,) and divide that pro- duct by 11, to ascertain how many times -y- is contained, which gives 1225 rods, and 3 half-yards remaining. Since it takes 40 rods for 1 furlong, there will be ^ as many fur- longs as rods ; ^j^ of 1225 are 30 furlongs, and 25 rods re- maining. And, since it takes 8 furlongs for 1 mile, there will be ^ as many miles as furlongs ; -|- of 30 are 3 miles, and 6 furlongs remaining : making, in all, 3 miles, 6 furlongs, 25 rods, 2 yards, and 1 foot, which is the answer required. The 2 yards are obtained thus : 1 of the 2 feet, and the 6 inches, make \ yard, which, with the f yards, make f = 2 yards. 308. Reduction defined. Reduction is the changing of a compound number mto a simple number of the same value, as in the first example, (JOTj) or the changing of a simple number into a compound number of the same value, as in the second example ; or, the 1^ ARITHMETIC. changing of a nurriber of any kind into another of the same value, as there are frequent examples in this book. S09. Observation. Observe, (Q07^) thatj in the first example, the simple number of the highest denomination ^7^ the given compound number, is reduced to the next lower denomination, to which is added what there may be of this loioer denomination ; that this sum is reduced to the next lower denomination still, and increased as before, and so on, till all is reduced as loio as desired : the reduction in each case being performed by mul- tiplying by the number which expresses how many units of the next lower denominMion make a unit of the simple number to be reduced* Observe, also, that, in the second example, the number in each denomination is divided by the number which expresses how many units of its own denomination make a unit of the next higher derwmination ; that the last quotient, together with the several remainders, form the compound number required, 310* Exercises in the Keduction of Compound Num- bers. Solve and explain the following problems, on the left, like the first, and those on the right, like the second, of the above examples (207). Long Measure. 1. Reduce 6 m. 3 fur. 20 rds. 3 yds. and 2 ft. to feet. 3. How many rods in 25 miles ? 2. Reduce 34001 feet to higher denominations. 4. How many miles in 16000 rods ? 1 . Reduce 27 yds. 1 qr. and 3 n. to nails. 3. What will 54 yds. 3 qrs. of cloth cost, at $.25 per yard ? 311. Cloth Measure. 2. Reduce 3627 inches to higher denominations. 4. How much cloth may be bought for $75.25, at $.25 per nail ? 313. Square Measure. 1. How many rods in a pasture which measures 2 m. 320 acres, and 80 rods ? 2. Reduce 1548800 yards to acres. COMPOUND NUMBERS. 143 3. What will 1 acre and 20 rods of land cost, at $.05 pe r foot ? 4. How much land may be bought for $1575.25, at $.05 per foot ? 213 Cubic Measure. 1. Reduce 4 yds. 15 ft. and 144 inches to inches. 3. How many inches in 3 toils of timber ? 5. What cost 5 cords and 3 cord-feet of wood, at $.75 per cord-foot ? 2. Eeduce 81648 inches to higher denominations. 4. How much timber in 267840 inches ? 6. How much wood may be bought for $79.20, at $.75 a cord-foot ? 314. Dry Measure. 1. Reduce 5bu. 3pks. and 1 iial. to quarts. 3, What cost 2bu. Ipk. ar d 3 qts. of chestnuts, at $.04 per quart ? 2. How many bushels in 10752 cubic inches ? 4. What would be the price per bushel, if $60 be paid for 1920 pints ef shag- barks ? SltS* Liquid Measure. 1. How many pints in 25 gals. 3 qts. ? 3. What would a milk- man receive for 300 cans of milk, each holding 2 gals, and 2 qts., at $.05 per quart ? to 2. Reduce 1732 gills higher denominations. 4. How many gallons of molasses in a cask gauging 7238 cubic inches ? 216. Troy Weight. 1. In 7 lb. 11 oz. 3dwt. 9 grs. how many grains ? 3. What will lib. 10 oz. 15 dwt. 20 grs. of jewelry cost, at $.04 per grain ? 2. Reduce 45681 grains to- higher denominations. 4. How much jewelry may be bought for $975.20, at $.02 per grain ? 217. Apothecaries' Weight. 1. In lib. 7, 25, 19, 12 I 2. Reduce 9876 grains to gis. how many grains ? | higher denominations. 144 ARITHMETIC. 3. Reduce 61b. lOg, 75, 29, IGgrs., to grains. 4. In 39S36 grains, how many pounds ? 218. Avoirdupois Weight. 1. Reduce 2 tons, 1200 lbs. 13 oz., to ounces. 3. What cost 20 tons, 500 lbs. of hay, at $.0075 per pound ? 2. How many tons, &c., in 1539000 drams ? 4. How much hay may be bought for $34.05, at $.005 per pound ? 219. Time. 1. How many seconds old IS a boy who has lived 12 y. 90 d. 15 h. 20 m. 30 s. ? 3. If a clock tick 60 times a minute, how many times would it tick in 16 years ? 2. How many days has a child lived, whose age is 31536000 seconds? 4. How long has a watch run, whose minute hand has turned round 46975 times ? 290. Circular Measure. 1. Reduce 1 sign, 15 to seconds. 3. If Massachusetts ex- tends 3 4r in longitude, what is its extent in seconds ? 2. Reduce 749408 seconds to higher denominations. 4. What is the extent of Massachusetts in latitude, it being 5940 seconds from south to north ? 221. English Money. 1. Reduce 25 10s. 6d. 3qrs. to farthings. 3. In 125 15s. Canada currency, how much Federal to 2. Reduce 9750qrs higher denominations. 4. In 9600qrs. N. Y. cur- rency, how much Federal money ? money f 222. Model of a Recitation. 1. Reduce ^^ to farthings. There will be 20 tA vtSt4 = ^%^ ^^s* = ^^^ 9[^s* times as many shil- lings as pounds, or f-H^' > 12 times as many pence as shillings, or ^^^. ; and COMPOUND MUMBERS. > 145 4 imes as many farthings as pence, or ^^^ qrs. = 166| qrs., wliich is the answer required. 2. Reduce y2ir to shillings, pence, and farthings. Perform the multiplications by dividing S6)83(2s. the denominator, (IO65) or divisor, when 72 practicable. Thus : 3)TI(3d. There will be 20 times as many shillings 9 as pounds, or T(T=2-Ty = ft = ^^s. -J-^ -Q shillings will make 12 times as many pence, 4 ii=T^ -V- = ^i^- P^^^y ^i^l ^^^^ ^ qTq/02 nro ^'^^^^ ^^ ^^^y farthings, f^ = .| = 2| qrs. ^)i^t qrs. j^ ^^^^ 2s. 3d. 2|qrs., the answer required. 6 "2 2 "0 3^I3. EXEECISES IN REDUCING A FRACTION TO UNITS OP LOWER DENOMINATIONS. In like Trmmver^ solve and explain the following problems, 1. Reduce t^tj ^^ *^^ fraction of a farthing. 2. How many shillings and pence in | of a pound ? 3. Reduce -^^^-^ of a mile in length to the fraction of a rod. 4. What is the value of f- of a mile ? 5. What fraction of a rod is yyV"? ^^ ^^ ^^^ 6. What is the value of |^ of an acre ? 7. Reduce ^^^ lb. Troy to the fraction of an ounce. 8. Reduce |^ of a Troy pound to ounces, pennyweights, and grains. 9. What fraction of a grain is ^^^ of an ounce. Apotheca- ries' weight ? 10. Reduce /^S ^^ drams, scruples, and grains. 11. Reduce ^f^ of a pound. Avoirdupois weight, to the fraction of a dram. 12. Reduce fy of a ton to lower denominations. 13. What fraction of a quart is yf^ of a bushel ? 14. Reduce |- of a bushel to quarts. 15. Reduce 2-^ of a gallon to gills. 16. How many quarts, pints, &c., in ^ of a gallon? 17. How many cubic inches in ^ sizB of a yard ? 18. Reduce -^ of a cubic yard to inches. 13 146 ARITHBIETIC. 19 What fraction of a day is y/ft ^f a year ? 20. What is the value of ^jj of a day ? 21. Reduce -^^ of a degree to seconds. 22 Reduce J of a circle to lower denominations. 3^4* Model of a Recitation. 1. What part of a pound is ^^Q- qrs. ? There will be j^ as many pence as ^^aitgfe^iV?- farthings, or \o^od. (114,) y'^ as many shillings as pence, or ^fjs., and 2V ^s many pounds as shillings, or Xyjj-, the answer required. 2. What part of a pound is 4s. 6d. ? There will be 12 times as many pence as shillings, and 6 added, giving 54d., and there will be ^^q as many pounds as pence, which (86) gives ^^^jj =^ -^j as required. 3SS. Exercises in reducing lower Denominations to THE Fraction of a higher. I71 like manner, solve and explain the following problems, 1. Reduce f d. to the fraction of a pound. 2. What part of a pound is 10s. 6d. ? 3. Reduce |- of a rod in length to the fraction of a mile, 4. Reduce 6 fur., 26 rods, 11 feet, to the fraction of a mile. 5. What fraction of an acre is t^tj of a rod ? 6. Reduce 3 roods, 13 rods, 90 feet, to the fraction of an acre. 7. What part of a Troy pound is |- of an ounce ? 8. Reduce 7 ounces, 4 dwt., to the fraction of a Troy pound. 9. What fraction of a pound is ^ of a grain. Apothecaries weight ? 10.' Reduce |^9 to the fraction of an ounce. 11. What fraction of an ounce is 35 29, 10 grs. ? 12. Reduce | of a pound to the fraction of a ton. 13. Reduce 1000 lb. 12 oz. 12 dr. to the fraction of a ton. 14. What fraction of a bushel is | of a pint ? 15. Reduce 3 pks. 1 gallon to bushels. 16. Reduce J|| of a gill to gallons. 17. What part of a gallon is 1 qt., 1 pt., 1 gill ? COMPOUND NUMBERS. 147 18. How many yards in -^ o( a, cubic foot ? 19. Reduce 81 cubic inches to yards. 20:, What part of a year is 24 hours ? 21. Reduce 25 minutes, 30 seconds to days. 22. Reduce of a minute to degrees. 23. What part of a circle is 4 signs, 15 degrees, 15 ninutes, and 15 seconds? SSO. Model of a Recitation. 1. Reduce .6 (183) of a bushel to lower denominations. .6 bushels. There will be 4 times as many pecks as 4 bushels, or 2.6 pecks ; the fraction .6 pecks, 2.6 pecks. will make 2 times as many, or 1.3 gallons ; 2 the fraction .3 gal. will make 4 times as 1.3 gallons. many, or 1.3 quarts ; in all, 2 pecks, 1 gal. 4 1.3 qts., which is the answer required. 1.3 quarts. SJ27. Exercises in reducing Decimal Fkactions to units OF A lower Denomination. In like manner , solve and explain the following problems, 1. What is the value of .625 of a bushel ? 2. Reduce .3125 of a mile to lower denominations. 3. Reduce .83 yards to lower denominations. 4. Reduce .375 acres to lower denominations. 5. How many cubic feet in .1875 of a cord? 6. How many quarts, pints, &c., in .6 of a gallon ? 7. What is the value of .875. ? 8. What is the value of .4 of a ton ? 9. Reduce .75 of a year to units of lower denominations. 10. If the moon advance in its orbit .406779661 + of a sign in a day, what is its daily advance ? 11. Reduce .625 to shillings and pence. 2S8. Model Sf a Recitation. 1. Reduce 2 pks. 1 gal. 1.3 qts. *to the decimal of a bushel. 148 AEITHMETIC. The 1.5 quarts will make ^ as many 1.3 quarts. gallons, that is, .3 gallons, (ISS,) which 1.3 gallons. annexed to the 1 gallon, make 1.3 gallons; 2.6 pecks. 1.3 gallons will make | as many, or .6 X"j r~j pecks, which annexed to the 2 pecks make 2.6 pecks ; 2.6 pecks will make ^ as many, or .6 bushels, which is the answer required. Or^ 2pks. 1 gal, 1.3 qts., equal to 21.3 qts., which will make -r}^ as many bushels, or s^j-^ of a bushel, (9245 2,) this common fraction reduced to a decimal fraction, (ITS^) makes .6 of a bushel as before. 230* Exercises in reducing lower Denominations to the Decimal of a higher. In like manner^ solve and explain the following prohleTns, 1. Reduce 1 pk. 1 gal. 1 qt. 1 pt. to the decimal of a bushel. 2. Reduce 3 fur. 15 rods, 2 yds. to the decimal of a mile. 3. Reduce 1 qr. 3 n. 2 in. to the decimal of a yard. 4. Reduce 25 yds. 3 ft. 72 inches to the decimal of a square rod. 5. Reduce 10 ft. 144 inches to the decimal of a ton of timber. 6. What part of a gallon is 1 qt. 1 pt. 3 gills ? 7. What part of a pound, Troy, is 1 oz. 1 dwt. ? 8. What part of a ton is 15001bs ? 9. Reduce January to the decimal of a year. 10. WTiat part of a revolution will a person's shadow make in 1 hour, if its hourly motion be 15 degrees? 11. Reduce 10s. 6d. 3qr. to the decimal of a pound. 330. Illustration of the Mode of reducing English Money by Inspection. Shillings, pence, and farthings, may be reduced to the decimal of a pound ; or, a decimal of a pound, to shillings, pence, and farthings, more expeditiously by inspection, as follows. 1. Reduce 15s. 7d. 3qrs., to the decimal of a pound. Since 20s. = 1; 2s. = 14s. =.7 /^ = ^i^ = .l; therefore, Is. =.05 write .1 for every 2 shil- 7d. 3qrs. = .032 -f- lings, or 2 shillings for every .1. Thus, 14 of the 15s. 155. 7d. 3qrs. = .782 + will make .7. vr COMPOUND NUMBERS. 149 Since ls. = ^ = -j-^^ =:.05, write .05 for an odd shilling, or 1 shilling for .05. Thus, 15s. make .75. Since 24qrs. = ^^(s = .025, add 1 to the farthings for e\ery 24 farthings in the given pence and farthings, and they w 11 become thousandths of a pound ; or subtract 1 from the thousandths of a pound for'every 25 thousandths in the given I thousandths, and the remainder wilLbe farthings. Thus,7d. I 3(^rs. = 31qrs,, which will be a trifle more than .032 ; in f all, .782 -f-- whi'^.h is the answer required. 2J1. Model of a RecitatioxN. ]. Reduce .782 to units of lower denominations. Since .1 is 2 shillings, {^3^. .7 = 14s. ,7 will bR .14 shillino-s. ai>dtOi5 .05 = Is. being 1 shilling, .75 will be 15 .032j= 7d. 3qrs. shillings; also, .025 being S4 7.g9-g i.5g 7(i 3qrs, farthings, or 1 subtracted from every 25 thousandths leaving far- things^ the .032 will be a trifle less than 31qrs. =:7d 3(irs. ; in all, 15s. 7d. 3qrs., which is the answer required. 93S. Observation. Observe, (230,) that it will be^sujficiently accurate, in reducing farthings to thousandths of a pound, to add 1 to the farthings if they are more than one half of 24, or 2 if they are more than \\ times 24 ; and, in reducing thousandths of a pound to farthings, (231 ,) to subtract 1 from the thou- sandths if they are more than one half of 25 ; or 2, if they are more than \\ times 25; siiice in either case the error will be less than one half of a farthing. 233. Exercises in reducing English Money by In- spection. In like manner, solve and explain the following problems. 1. What is the value of .25 ? 2. Express 6s. 6d. in a decimal. y^-^ 3. Reduce .625 to shillings and pence. ^ 4. Reduce 7s. 6d. to pounds. 5. Reduce .875 to lower denominations. 6 Reduce 4s. 5d. 3qrs. to the decimal of a pound. 7. Reduce .323 to units of lower denominations. 13* j* / i I' 150 ARITH31ETIC. 8. Express 5s. 4d. in a decimal of a pound. 9. What is the value of .116 ? 10. Express 4|d. in pounds. 11. What is the value of .075. 12. Express 19s. 6d. 1 qr. in pounds. 13. What is the value of .48 ? 14. Express 12s. lid., 3 qrs. in pounds. 15. Reduce .1875 to shillings, &c. 16. Reduce Is. 2d. 3 qrs. to pounds. 17. Reduce .6 to shillings, &c. 18. Rpr!^7rr, os. Su= CO poimds. 19. Reduce 12.18 to pounds, shillings, pence, and fur 'educe 125 lis. 3d. to the decimal of a pound. 934 Model of a Recitation. 1. If from a piece of broad-cloth, measuring 35 yds. 3 nls., a tailor cut 6 yds. 1 qr. for a cloak ; 3 yds. 3 qrs. 1 n. for a surtout ; 2 yds. 2 qrs. 2 nls. for a frock-coat ; 1 yd. 2 qrs. for a pair of pantaloons ; and 2 qrs. 2 nls. for a vest ; how much cloth would he use for these garments ? To ^ascertain how much cloth he would use, you must add together the several quantities cut off from the piece. Arrange together the numbers to be added so that the simple numbers of each denomi- nation may stand in a column by themselves. Add the numbers of each denomination separately, beginning with the lowest. The 14 3 1 5 nails of the first column are equal to 1 nail, which write in its own column, and 1 quarter, (lO^j) vvhich add with the other quarters, making 11 quarters, equal to 3 quarters, *which write in their own column, and 2 yards, which add with the other yards, making 14 yards, which write in their own column, making in all, 14 yds. 3 qrs. 1 nl., which is the answer required. S33. Observation. Observe, (SSAj) that^ in the addition of compound iiuvi- hers, the amount of each denomination must be reduced to units of the next higher denomination, a?id added there, a?id yds. qrs. nia 6 1 3 3 1 2 2 2 1 2 2 2 I COMPOUND NUMBERS. 151 tlat in each column^ only the excess over exact units of the mxt higher denomination are to be written. 236. Exercises in adding Compound Numbers. In like manner^ solve and explain the following problems, 1. How much cloth in 4 pieces, measuring as follows : 25 yds. 3 qrs. ; 27 yds. 1 qr. 2 nls; 30 yds. 3 nls ; and 28 yards? 2. How far would a horse trot in 5 hours, if he should bot the first hour 12 miles, 3 fur. 25 rods ; the second hour, 1 1 miles, 7 fur. 20 rods ; the third hour, 12 miles, 5 fur. 36 r)ds, 5 yds.; the fourth hour, 11 miles, 6 fur. ; and the fifth liour, 13 miles, 15 rods? 3. If by one road, from Lowell to Boston, the distance be 25 m. 2 fur. and 20 rods, and by another road, the distance be 24 m. 7 fur. 12 rods ; how much distance is traveled in rid- ing to Boston by one road and returning by the other ? 4. How much land in a farm which consists of 50 acres, 2 roods, 33 rods, wood-land ; 25 acres, 14 rods, mowing-land ; 80 acres, tillage ; 20 acres, pasturing ; 12 acres, 1 rood, covered with water ; and 10 acres, 25 rods, swamp ? 5. How much timber in two sticks, one of which measures 2 tons, 20 feet, 1642 inches ; the other, 1 ton, 15 feet, 1295 mches ? 6. How much wheat does that man raise^ who has three, fields, and raises on the first, 45 bush. 3 pks. ; on the second, 36 bush. 1 pk. 7 qts. ; and on the third, 30 bush. 2 pks. 1 quart? 7. How much molasses in two casks containing as follows : 70 gals. 3 qts., and 126 gals. 1 quart? 8. If a Johannes weigh 18 dwts. a doubloon 16 dwts. 21 grs., a moidore 6 dwts. 18 grs. and an English guinea 5 dwt. 6 grs. ; what is the weight of them all ? 9. If an apothecary mix of one kind, 7, 55? 29 ; of another kind, 2, 35 ; and, of a third kind, 29, 10 g/s. ; what is the weight of the mixture ? 10. If a load of hay weigh, without the wagon, 1 ton, 1200 lbs., and the weight of the wagon is 1984 lbs. ; what is the weight of the whole ? 11. If the load of hay mentioned in tiie last problem, were drawn over a bridge by two oxen and a horse, the oxen weighing 1 ton, 187 lbs., the horse weighing 1160 lbs., and the 152 ARITHMETIC. driver 165 lbs. 12 oz. ; how much more did the bridge sustain from tills team passing over it ? 12. If 15 14s. 6d. be paid for a pair of oxen, 14 for a horse, and 6 9s. 3d. for a cow ; what would be the whole cost? 13. How old would a man be when his eldest child is 12 years, 25 days, and 16 hours old, if he was 25 years, 344 days, and 10 hours old, at the birth of this son ? 337. Model of a Eecitation. How much cloth would remain, if 14 yds. 3 qrs. 1 nl. be cut from a piece measuring 35 yds. 3 nails ? To ascertain how much cloth would yds. qrs. nis. remain, you must subtract the sum of what 85 3 he used from the whole piece. 14 3 1 Write the subtrahend under the minuend, placing the simple numbers of each denomi- 20 1 2 nation under those of the same denomination. Beginning with the lowest denomination, take 1 nail from 3 nails, and 2 nails remain, which write in their own column ; reduce 1 of the 35 yards to quarters, making 4 quarters, (IO45) from which take the 3 quarters, and 1 quarter remains, which write in its own column, and take 14 yards from 34 yards, and 20 yards remain, which write in their own column, making in all, 20 yds. 1 qr. 2 nls., which is the answer required. S38. Observation. * Observe, that, in subtraction of compound numhersy luhen a number of any denomination in the minuend is less than the corresponding number in the subtrahend, a unit, or in some cases, a part of a unit, of a higher denomination in the minuend must be reduced to make up the deficiency, S30. Exercises in subtracting Compound Numbers. In like manner, solve and explain the folloioing problems, 1. If 5 yds. 1 qr. 3 nls. be cut from a piece of cloth measuring 20 yds. 3 qrs., how much would remain? 2. What is the difference between two piles of wood, one of which measures 15 cords, 3 cord-feet, 12 feet, and the other, 10 cords, 7 cord-feet, 6 feet ? COMPOUND NUMBERS. 153 3. If a farmer raise on one field 150 bush, of potatoes, and or another, 90 bush. 3 pks. ; how much more does he raise or the large field than on the other. 4. If a merchant draw from a cask of molasses, containing 16 gals. I qt. ; at one time, 13 gals. 3 qts. ; at another, 1( gals. ; and at a third time, 25 gals. 3 qts. ; how much would remain in the cask? 5. How much more does a Johannes weigh than a doub- lom? (aae^s.) 6. If for a horse worth 18 10s. a man should give a cow worth 8 7s. 6d., and a calf worth 1 16s. 4d., and the rest in money ; how much money would it require ? 7. How much quicker could a person travel from Lowell to Boston, in a car, than in a stage, if it should take the car 1 h. 20 m. 30 sec, and the stage 4 h. 15 m. and 45 sec. in ths passage ? 8. How much longer is the day than the night, when the sun rises 56 minutes past 4 o'clock, and sets 4 minutes past 7 o'clock ? 9. What is the difference of latitude between Boston ar d Cape Horn, Boston being 42 28' north, and Cape Horn being 55^ 2' south latitude ? and how much farther from the Equator is Cape Horn than Boston ? 10. What is the third angle of a triangle, (4395 14,) if the three angles equal 180, the first 44 13' 24", and the second, 79 46' 38"? 340. Model of a Recitation. 1. How much silver would be required for 15 spoons, if 1 oz. 14 dwt. 12 grs. be put into each spoon ? It would require for 15 spoons, 15 times lbs. oz. dwt. grs. as much as for 1 spoon. 15 times 12 grs. 1 14 12 is 180 grs., equal to 12 grs., which write 15 in their own column, and 7 dwts., (lOOj) which add with 15 times 14 dwts. making 2 1 17 12 217 dwts., equal to 17 dwts., which write in their own column, and 10 oz., which add with 15 times 1 oz., making 25 oz., equal to 1 oz., which write in its own column, and 2 lbs., which write in their own place, making in all, 2 lbs. lt)z. 17 dwts. 12 grs., which is the answer required. 154 arithmetic. 341. Exercises in multiplying Compound Numbers. In like manner^ solve and explain the folloioing problems. The contractions practised in the muUiplication and divisk)n of simple numbers, may be adopted here, whenever found more convenient. 1. If a silver thimble weigh 12 dwt. 12 grs., what would be the weight of 25 thimbles ? 2. If Lowell railroad be 25 m. 5 fur. 30 rods in length, how far would a locomotive run on this road in June, if it per- form 3 trips per day ? 3. How much cloth in 35 pieces, each piece containing 27 yds. 2 qrs. 3 nails ? 4. How much land in a man's farm which is fenced into 9 fields, each containing 3 acres, 2 roods, 25 rods ? 5. How much gravel could a man remove in 18 loads, at 1 yd. 25 feet each ? 6. How much would that cask hold, which could be filled with 35 pailfuls, each pailful being 9 qts. 1 pt. 2 gills ? 7. How many bushels of wheat in 135 bags, each contain- ing 2 bush. 3 pks. ? 8. What would be the weight of a box of 115 pills, if each pill should weigh 19, 4 grs. 9. What would be the weight of 15 barrels of flour, at 196 lbs. each ? 10. How much sterling money in $25, if 1 dollar make 4s. 6d. ? 11. If a person rise 1 h. 20 m. later than he ought to every morning for 12 years, how much time would be thus wasted ? 12. If the sun appear to move 15 per hour, what is its apparent motion in a day of 15.1 hours in length ? 13. Multiply 5 oz. 10 dwts. 15 grs. by 10. 14. Multiply 40 m. 3 fur. 25 rods by 15. 15. Multiply- 34 yds. 1 qr. 1 n. by 64. 16. Multiply 57 acres 3 roods 20 rods by 100. 17. Multiply 17 tons 40 ft. 512 in. by 500. 18. Multiply 50 gals. 2 qts. 1 pt. 1 gill by 25. 19. Multiply 84 15s. 3d. 2 qr. by 12. 20. Multiply 7 tons 1800 lbs. 12 oz. by 2.5. 21. Multiply 5 y. 212 d. 10 h. 15 m. by 100. COMPOUND NUMBERS. 155 S'US* Model of a Recitation. 1. If 2 lb. 1 oz. 17 dwt. 12 grs. of silver be put into It spoons, what would be the weight of each spoon ? lbs. oz. dwts. grs. Each spoon would weigh -^ as 2 1 17 12 much as 15 spoons. As 15 is not 12 contained in the 2 lbs. reduce them to ounces, making with the 1 oz. It) 25 25 ounces, which divided by 15 - gives 1 oz. for the quotient, and 1 oz. 10 10 oz. remainder, which reduced 20 to pennyweights, make with the 17 * dwts. 217 dwts., which divided by 15) 217 15 gives 14 dwts. for the quotient, and 7 dwts. remainder, which re- 14 dwts. 7 duced to grains, make with the 12 24 grs. 180 grs., which divided by 15 gives 12 grains for the quotient, 15)180 making in all, 1 oz. 14 dwts. 12 grs., which is the answer re- 1 oz. 14 dwts. 12 grs. quired. 3^t3. Exercises in dividing Compound Numbers. In like manner, solve and explain the following problems. 1. What would be the weight of 1 dollar, the weight of 8 dollars being 6 oz. 18 dwts. 16 grs. ? 2. How far does that boy live from his school-house, who has to travel 170 m. 2 fur. in attending school twice a day, for 60 days ? 3. How long is that room which requires 27 yds. 2 qrs. of carpeting cut into 5 pieces, to carpet it ? 4. If 135 acres be fenced off into 16 equal lots, what would be the size of each lot ? 5. If a man team to market 25 cords, 5 cord-feet of wood, at 20 loads ; how much would that be a load ? 6. What is the contents of each of such bottles that 160 of them could be filled from a cask holding 115 gallons ? 156 i^RITHMETIC. 7. How many apples in a barrel, if 101 bush. 1 pk. make 45 barrels ? 9. What would be the weight of a dose of medicine, if 4 g, 5 5, 2 9, 12 grs. be taken at 12 doses ? 9. If 33 lbs. of steel be put into 256 axes, how much would. that be apiece ? 10. If 147 bushels cost 47. 12s. 5d., what does it cost per bushel ? 11. If a teacher devote 5 hrs. 30 min. per day to 50 scholars, how much would that be for each scholar? 12. What would be the daily motion of the moon, if it complete a revolution in 29| days ? S4J:. Model of a Recitation. How many rods round a pasture, measuring on the first side ^ of a mile, on the second 1.17 furlongs, on the third 2^ furlongs, and on the fourth 1 furlong 18.8 rods, ^^m. =jJL>^|;i^ rods =110 rods. 1.17 fur. =1.17 X 40 rods = 46.8 " 2/^ fur. =2/^X40 rods =: 86 1 fur. 18.8 rods = 40 rods + 18.8 rods = 58.8 " 301.6 rods. These quantities being expressed in different ways, mvj^t^ before they can be added, he reduced to numbers of the same, kind. Multiply ^^ m. by 8 to reduce it to furlongs, and that pro- duct, also, 1.17 furlongs, 2^g- furlongs, and 1 furlong, by 40 to reduce them to rods ; these several quantities, being made alike and added, make 301.6 rods, which is the answer required. S4l^* Exercises in the use of Numbers variously EXPRESSED. In like mannei , solve and explain the following problems, 1. How much is -f^ of a week and J of a day ? 2. What is the difference between two fields, one of which measures 2|- acres, the other 1 acre 2|- roods ? 3. How much cloth in three remnants, tbe first measuring 2.4 yards, the second 1 yd. 3 qrs., and the third f of a yard ^ 4. How many cubic inches in 4 bushels, 1.375 bushels and ^ of a peck ? COMPOUND NUMBERS. 157 5. If 2| pecks be taken from a bag holding 2.75 bushels ; ho .y much would be left ? 3. How much water in a pail measuring 10.75 quarts, but wanting -^^ of a gallon of being full ? 7. How much silver in a large and small spoon, sugar- tor gs, and butter-knife, weighing severally, 2.9 oz., 1 oz. 12|. dwt.. If oz. and 18.5 dwt.? B. If an apothecary should mix a medicine at a cost of $A3 per ounce, and should sell it at $.48 per ounce avoirdu- pos; how much would he gain in selling 10 lbs.? ). Add -^ of a ton, 16^^ lbs., ^\ of a ton, and .83 of a tDn together. 10. Subtract | of a shilling from 1.25. 11. What is the difference between 52 wks. 1 d. 6hrs. and 365.25 days? - 12. If two ships sail from the same point, one north IB/^ degrees, the other south 25 33^' ; what then would be the lat tude between them ? S416. Model of a Recitation. How many years, months and days from the first resistance with arms in the American revolution, April 19th, 1775, to the declaration of Independence, July 4th, 1776? To answer this question you must subtract the time between the Christian era and the earlier 1775 ^6 3^ date, which is 1774 years, 3 months and 1774 3 18 -^^ days, from the time between the Christian era and the later date, which is 1 2 15 1775 years, 6 months and 3 days, the hours, &c., being disregarded. Reduce 1 of the 6 months to days, making, (20^5 ) with the 3 days, 33 days, from which 18 being subtracted, 15 days remain ; 3 months from the other 5 months leave 2 months ; and 4 years from 5 years leave 1 year, making, in all, 1 year, 2 months and 15 days, which is the answer required. But a more convenient way of obtaining the same result is, instead of writing the number of years, 1776 7 4 months and days, to write the order of 1775 4 19 the year, month and day, that is, the dates themselves. Thus, from the 1776th 1 2 15 year, 7th month and 4tn day, subtract the 1775th year, 4th month and 19th day ; 14 j5S arithmetic. precisely as so many years, months and days. This increases each number in each denomination by the same quantity, 1, and, consequently, does not affect the difference. 24:T Exercises in finding the Difference of Time BETWEEN Dates. In like manner, solve and explain the following problems, 1. How long from the time that Washington entered up/^n the command of the American army at Cambridge, July '2d, 1775, till the disbanding of the army at West Point, Novem- ber 3d, 1783 ? 2. How long was General Harrison's victory at Tippecanoe, November 7th, 1811, before General Jackson's victory at New Orleans, January Sth, 1815 ? 3. How long from the date of a note. May 10th, 1835, till its payment, June 5th, 1840 ? 4. How old is that man, June 27th, 1840, who was born March 23d, 1807 ? 5. What is the date of that note, which was paid Decem- ber 31st, 1839, 2 y. 3 m. 11 d. after its date? 6.' When was that note paid, which was dated August 4th, 1836, and paid 3 y. 3 m. 30 d. after date ? 7. Yiow much older is Lizzie, born Sept. 11th, 1843, than Mary, born April 15ih, 1846 ? 8. How long was John absent, having left town July 1st, 1839, and returned August 25th, 1840? 248. Model of a Recitation. What cost 4 bu< 'a. 3 pks. 1 gal. of wheat, at 5s. 6d. per bushel ? The whole cost will be the product of the price of one bushel by the number of bushels; 4.875 bushels. but, before they can be multipliea .275 . together, they must be reduced to ^ simple numbers ; 4 bush. 3 pks. 24375 1 gal. reduced to bushels is 4.875 34125 bushels, (^SS,) and 5s. 6d. reduced 9750 to pounds is .275 (230.) Now, since 1 bushel costs .275, the whole 1.340625= cost will be .275 as many pounds 1. 6s.9|d. as bushels, which is 1.340625 equal (231) to 1. 6s. 9|d. COMPOUND NUMBERS. 159 S^9. Exercises in the Reduction of Compound Num- bers FOR Multiplication. In like manner^ solve and explain the following problems. 1. What is the value of 15 acres, 2 roods, 20 rods, $62.25 being the cost of each acre ? 2. What would 12 miles, 3 furlongs, 32 rods of road cost, at 175. 10s. 6d. per mile? 3. Goliah, measuring 6 J cubits of 1 ft. 7.168 in. in height, was how tall in feet and inches ? 4. What is the cost of 5 yds. 1 qr., 2 nls. of broadcloth, at $5.50 per yard ? 5. What is the value of 2^ tons, 1| tons, and 2| tons of bay, at $12.25 per ton? 6. What will 4f tons of iron come to, at 20. 15s. 6d. per tm? 7. What will 8f hogsheads of molasses, at 63 gallons each, come to, at 2s. 6d. per gallon ? 8. At 5s. per bushel, what will 4 bush. 2 pks. 1 qt, of corn come to ? 9. Bought a silver cup, weighing 9 oz. 4 dwt. 16 grs., at Cs. 8d. per ounce, what was the whole cost? S^SO. Model of a Recitation. How many square feet in a square, measuring 16 ft. 6 in. on each side ? 16.5 16.5 Since the contents is the product of the length by the breadth, (lOS,) and 16 feet 825 6 inches being 16.5 feet, the contents will 990 be 16.5 X 16.5 = 272.25 square feet, which 165 is the answer required, (4395 ^^) 272.25 feet. 3S1* Exercises in the Mensuration of Surfaces and Solids. In like manner, solve and explain the following prohleTns, 1. How many square inches (105) on the page of a book 8 inches long and 5 inches wide ? 2. How many square yards in a square, measuring 5 J yards on each side ? 160 ^ AHITHMETIC. 3. How many feet in a floor which is 16| feet long and 15 feet wide ? 4. How many square yards will carpet a floor which is 5 yds. 1 ft. 6 in. long, and 5 yards wide ? 5. How many rods in a garden 5 rods, 2| yards long, and 4.5 rods wide ? 6. How much land in a field 26 rods, 11 feet long, and 6 rods wide ? 7. How many feet in a board 17 ft. 9 in. long, and 1 ft. 6 in. wide ? 8. If from a square stick of timber 1 foot wide and 1 foot thick, you saw off a piece 1 foot long, that block would contain exactly 1 cubic foot; how many cubic feet in such a stick of timber 16 feet long ? 9. How many boards 1 inch thick could be made of that stick, allowing no waste in sawing ? 10. How many cubic inches in one of the boards ? 11. How many cubic inches in all of the boards ? 12. How many cubic inches in a stick of timber 1 foot wide and thick, and 16 feet long ? 13. How many feet in a stick of timber 24 feet long, 1.8 feet wide, and 1.5 feet thick? 14. How many feet in 2 sticks of timber, each 36 feet long, 2 ft. 6 in. wide, and 2 ft. 3 in. thick ? 15. How many feet in a load of wood 8 ft. long, 3 ft. 6 in. wide, and 3 ft. 9 in. high ? 16. How many feet in a load of gravel 7 ft. 6 in. long, 4 ft. 3 in. wide, and 2 ft. 3 in. high? 17. How many yards of gravel must be removed to make a cellar 2.5 yards deep, 6 yards long, and 5.6 yards wide ? IS. How many yards of stone work in a wall 38| yards long, 4 ft. 6 in. high, and .8 of a yard thick? 19. How many feet in a room 17 ft. 6 in. long, 15 ft. 3 in. wide, and 10 ft. 9 in. high ? 20. How many cord-feet in a load of wood 8 feet long, 4 feet wide, and 4 feet high ? 21. How many cords of wood in a pile 32 feet long, 4 feet wide, and 7 feet high ? 333. Illustration of the mode of Abridging the Pro- cess OF solving Problems. 1. What is the value of a pile of wood 64 feet long, 4 feet wide, and 6 ft. 6 in. high, at $5.25 per cord ? COMIOUNP NUMBERS. 161 In questions like this, involving both multiplication and division, it will be most convenient, and will generally much abridge the process, to express all the operations before per- forming any of them. Thus; the length 64 feet muhi- plied by the 64 X 4 X 6.5 H- 16 ~ 8 X 5.25 = $68.25. breadth 4 ft. will give the s:iuare contents of the base, which multiplied by the height C ft. 6 in., or Q.5 feet, will give the cubic contents (190) ii feet; this divided by 16 will give the contents in cord-feet, and this quotient divided by 8 will give the con- t3nts in cords, which multiplied by the price of 1 cord, will give the whole value, or the answer required. The whole process being thus expressed and explained, jerform the operations indicated by the signs, in such order as v/ill require the fewest figures; thus, divide the 64 by the 16 ; tbe quotient, 4, multiply by the 4; the product, 16, divide hy the 8; multiply Q.3 by the quotient, 2, and multiply $5.25 by that product, 13, making $68.25, which is the answer required. Perhaps it will be more convenient still, to express the process in a fractional form, (865) by making the divisors factors of the denominator, and then to reduce the fraction to its lowest terms. ^ 2 (121.) Thus, the 16 ^^y^^ 64 in the numera- tor, may be canceled from both terms (ISl) ; and 4, the other factor of 64, multi- plied into the other 4 of the numerator, makes 16, of which the 8 in the denominator is a factor; consequently, 8 may be canceled from both terms ; and 2, the other factor of 16, mul- tiplied into Q,5 makes 13, which multiplied into $5.25 makes $68.25, as before. S^3* Exercises in solving Problems involving both Multiplication and Division. In like manner, solve and explain the folloioing problems. 1. How many cords of wood in a pile 40 feet long, 4 feet wide, and 9 ft. 3 in. high ? 14* 162 ARITHMETIC. 2. What is the value of a load of wood, measuring S feet in length, 4 ft. 6 in. in width, and 5 ft. 3 in. in height, at $8 per cord ? 3. What is the value of a stick of timber, measuring 50 feet in length, 2 ft. 6 in. in width and thickness, at $4 per ton? 4. What would be the cost of digging a cellar 19 ft. 6 in. long, 15 feet wide, and 10 ft. 6 in. deep, at $.25 per yard? 5. How many acres in a pasture 36 rds. 8.25 ft. long, and 30 rods wide ? 6. How many yards in a floor 28 ft. 9 in. long, and 22 ft. 4 in. wide ? 7. How many yards, in length, of carpeting, which is 4 ft. 6 in. wide, will cover a floor 17 feet long, and 15 ft. 6 in. wide ? 8. How many days would Samuel have to go to school, twice per day, to travel 1000 miles, if he live 5 furlongs from school ? 9. How many times would a wagon wheel, 13 ft. 9 in. in circumference, revolve in running 25 miles, 6 furlongs ? 10. How many times could a coal-basket, holding 1 bush. 1 gal. 2 qts., be filled from a coal-cart, containing 65 bush. 1 pk. 2 qts. ? 11. What would a.hogshea4 of cider, containing 62 gals. 2 qts. come to, at $.04 per bottle of 1 pt. 2 gills ? 12. What is the value of a lot of spoons, weighing 9 lbs. 10 oz. 4 dwt., each spoon weighing 16 dwt. 10 grs., and worth $1.25? 13. How many loads of hay, each weighing 1750 lbs., in a stack, weighing 16 tons, 875 lbs ? 14. How many pills may be made of a mixture of lOg 43, each weighing 13,10 grs. ? 15. In 67. lis. 7d., how many crowns, at 6s. 7d. each ? 16. How many yards of English cassimere, at 12s. 8d. per yard, may be bought for 395. 4s. sterling ? 17. What part of 1, or 20s. is 15s. ? 18. If a yard of broadcloth cost 17s. 6d. ; what part of a yard might be bought for 13s. 6d. ? 19. What part of 1 15s. 9d. is 15s. 9d.? COMPOUND NUMBERS. 163 9S4. Model of a Recitation Reduce 64 17s. 6d. of sterling money, and of Canada, !^"ew England, New York, Pennsylvania, and Georgia, cur- rencies to Federal money; and the results back again as bifore. i*fi4. R7^ Reduce 17s. 6d. to ^^ the decimal of a pound, (SSOj) and, since in 9)2595.000 ^'^'^""^ 7"^y \^J ' one pound equals $4|, $288.33^ there will be 4|, or ^ q^ times as many dollars as pounds, which gives 40 \ 2595 00 S288.33^. ' Since one dollar equals 64.875 = 64. 17s. 6cl. ^ ' '^^'^ f'" ^, f ^' many pounds as dollars ; which gives 64.875. 64.875 Since, in Canada currency, (SOS,) one 4 pound equals $4, there will be 4 times as many dollars as pounds ; which gives 4 ) $259.5000 $259.50. Since 1 dollar equals J, there will 64.875. be J as many pounds as dollars ; which gives 64.875. Since, in N. England currency, (SOS^) 3 ) 64.875 one pound equals $3 J , there will be 3 J , or ^^ times as many dollars as pounds; divide $216.25 by 3 to obtain |, and remove the point .3 one place farther to the right to obtain J# ; which gives $216.25. 64.875. Since $1 =? .3, there will be .3 as many pounds as dollars, or 64.875. Since, in New York currency, (SM>tl,) 4) 64.875 1 = $2|, there will be 21, or J^ times as many dollars as pounds ; which $162.18f gives $162.18|. .4 Since $1 = .4,' there will be .4 as many pounds as dollars; which gives 64.875. 64.875. 164 ARITHMETIC. 64.875 8 Since, in Pennsylvania currency, 3 ) 519.000 (SO^,) 1 = $2|, there will be 2|, or f times as many dollars as pounds ; $173. which gives $173. 3. Since SI == , there will be f as many pounds as dollars; which gives 8)519. 64.875. 64.875. 64.875 30 Smce, m Georgia currency, (90li.) 7 ) 1946.250 1 = ^f , there will be 42, or ^ times as many dollars as pounds ; which gives $278,034- $27S.03f 7 Since $1 = -^^, there will be -/^ as many pounds as dollars ; which gives 30 ) 1946.25 64.875. 64.875. 3^^. Exercises in- the Reduction of Currencies. In like wanner, solve and explain the following proh' lems. 1. One dollar is what part of a pound in sterling money, and in Canada, New England, New York, Pennsylvania, and Georgia currencies ? 2. What part of one dollar is one pound of sterling money, and of each of the currencies ? 3. How many dollars in 1 of sterling money, and of each of the currencies ? 4. Reduce 25 10s. from sterling to federal money. 5. Reduce $25,375 to sterling money. 6. How much Federal money would pay for a farm, in Canada, worth 500. 7. If a Canadian lumber merchant sell, in New Orleans lumber that cost him 875 for $3750, whether, and how much, would he gain, or lose ? COMPOUND NUMBERS. 166 8. If a farm, in Cambridge, Massachusetts, which, in 1740, cost 125 7s. 6d., be now worth $3000, how much -has it iiicreased in value ? 9. How many dollars should a New York merchant receive for 125 yards of flannel, at 3s. 6d. per yard? 10. If a wholesale dealer, in Philadelphia, receive $1500 fcr a quantity of cloth, at 3s. 9d. per yard ; how many yards ir . the quantity ? 11. If a Georgia planter sell his wheat at 3s. 6d. per bushel, and receive $387.50 ; how many bushels would he sell? 12. Where can the same kind of penknives be bought the clieapest, if 2s. 3d. apiece be the price ? 13. How much Federal money would a horse cost in each o:' the several currencies ; if the price be 20 18s. ? 14. Eeduce $.25 to each of the several currencies. 15. Reduce Is. 6d. of the several currencies to Federal ironey. Note. Whenever any of the denominations of English rrioney occur in the following pages, they will be in New E ngland currency, unless otherwise specified. 9!56 Model of a Recitation. 1. A merchant sold 3545 yards of cotton cloth, at 9d. per yard ; what was the amount of it in Federal money ? Since the price of 1 yard was |^ of a dollar, (SOOj) the amount of the whole quantity must have 8)3545 been -j- as many dollars as there were yards ; therefore, divide (93) the number $443,125. of yards by 8 to ascertain how ma^y dol- lars there were in the amount. 2. A merchant paid $35.31 J for cotton cloth, at 4Jd. per yard ; how many yards did he purchase ? Since the price of 1 yard was -^^ of a dollar, (SO65) he dUQCQii must have purchased 16 times as many * tfi y^^^^ ^^ ^^ P^^^ ^^^^^^^ (*^1) ' therefore, multiply the number of dollars that he paid t^nt- 1 by 16 to ascertain how many yards he purchased. 166 arithmetic. 3^7. Observation. Hence, observe, that,,(^i5^^) in the following questions, and lohenever the multiplier, or divisor, is such that it can be reduced to an aliquot part of a dollar, the process nmy he much abridged, by using the aliquot part. 958. Exercises in the Use of Aliquot Parts. In like manner, solve and explain the following problems. 1. What cost 4S72 oranges, at 3d. apiece ? 2. How many pounds of sugar, at 6d. per pound, may be purchased with $1^.58^ ? 3. How many dollars would it take to pay for 144 yards of calico, at Is. per yard ? 4. How many times is Is. 3d. contained in $5 ? 5. What is the value of 1728 bushels of apples, at Is. 6d. a bushel ? 6. How many bushels of potatoes would it take to come to $24.66|, at $.33J per bushel ? 7. What would be the cost of 24 yards of muslin, at 2s. 3d per yard ? 8. Bought 36 yards of bombazine, at 2s. 6d. per yard. What was the bill ? 9. How many pairs of half-hose, at $.45f a pair, may be bought for $11 ? 10. What would be the cost of 40 yards of linen, at 3s. per yard ? 11. Sold children's shoes, at 3s. 3d. a pair, to the amount of $54.16| ; how many pairs were sold ? 12. How many palm-leaf hats, at 3s. 6d. apiece, may be bought with $14.58J ? 13. What may I receive for 576 lbs. of wool, at $.62J a pound ? 14. How much flannel, at 4s. a yard, may be bought with $4.66 ? 15. If the expense of cultivating an acre of corn be $20, what would be the profits from a field of 12 acres, each yield- ing 50 bushels, worth 4s. 6d. per bushel ? 16. If a farmer sell his rye at $.83J per bushel, and receive $95 for it, how many bushels would he sell ? 17. How many days, at 5s. 3d. per day, must a man work to earn $63.87^ ? COMPOUND NUMBERS. 167 18. If a merchant sell 4 dozen pairs of gloves, at 5s. 6d. a pair, what would he receive for them ? 3il9. Model of a Recitation. At 7s. 6d. a bushel, what would 100 bushels of wheat come to? Since 7s. 6d., the 4.) $100 = cost at $1. per bushel. ?ji^\^Ui^'' wt^: J5 = cost at $_^ per bushel. ^^^f^^^^^ $125 = cost at $1.25, or 7s. 6d. per -^bertf ddlS bushel. ^u ... 1 m the cost at 1 dollar per bushel, arid J of the number of buShels would be the number of dol- lars in the cost, at J of a dollar per bushel ; therefore, write the number of bushels, and to it add J of itself, and the sum will be equal to the whole cost in dollars. S(&0. Exercises in multiplying by Units and Aliquot Parts. In like manner i solve and explain the following problems, 1. How much would 12 pairs of ladies' shoes come to, at 6s. 6d. a pair? 2. At 6s. 9d. a pair for silk hose, what would be the price per dozen ? 3. What would be the cost of 54 gallons of oil, at 7 shil- lings per gallon ? 4. How much would 5J pounds of green tea, at 8s. a pound, amount to ? 5. Bought 2.875 yards of satinet, at 8s. 3d. per yard ; what was the cost ? 6. How much would 11 pitchforks, at 9s. apiece, amount to? 7. Multiply 1840 by If 8. What would 12 weeks' work come to, at 10s. 6d. per day, Sundays excepted ? 9. How much would 2^ yards of cassimere cost, at lis. 3d. per yard ? 10. What would be the cost of 52 weeks' board, at 15s. per week ? 168 ARITHMETIC. 11. Paid for 18 weeks' board, at 13s. 6d. per week ; how much was the bill ? ^61. Illustration of the Principle of reducing Frac- tions BY Inspection. Since 100 cents make the unit, 1 dollar, any number of cents are so many hundredths of a unit ; thus, 12| cents is S.125. But 12 J cents is a ninepence, or ^ of a dollar (S06) ; therefore, the decimal for any number of eighths will be the number of cents in so many ninepences ; and, for any number of twenty-fourths, sixteenths, twelfths, and sixths, the decimal, in hundredths, will be the number of cents in so many threepences, fourpence-halfpennies, sixpences, and shil- lings, respectively. Also, any decimal, corresponding with the number of cents in any such part of a dollar, may be reduced to a common fraction, by writing the part instead of the decimal. 363. Model of a Recitation. 1. Reduce -f-^ to an equivalent decimal. Fourpence-halfpennies being sixteenths 5 *^19^ ^^ ^ dollar, the decimal for -^^ will be the ^^ * number of cents in 5 fourpence-halfpen- nies, which is 31J ; consequently, the decimal required is .3125. 2. Reduce .4183 to an equivalent common fraction. The hundredths, in this decimal, being A-icjf^ 5^ the same as the number of cents in 2s. 6d. T2 =.5 sixpences, and sixpences being twelfths (SOG) of a dollar, this decimal is equiva- lent to -^, which is the answer required. 363* Exercises in reducing Fractions by Inspection. In like manner , solve and explain the following problems, 1. Reduce ^V to a decimal. 2. Reduce .2083 to a common fraction. 3. What are the decimals equivalent to ^, ^^, and ^ ? 4. What are the common fractions equivalent to .7083, .7916, and .9583 ? r PROPOETION. 169 5. Reduce -j^ to a decimal. 6. Reduce .1875 to a common fraction. 7. What are the decimals equivalent to ^^, f^, and -f-J ? 8. What are the common fractions equivalent to .8125 and .^375 ? 9. Reduce -j^ to a decimal. 10. Reduce .583 to a common fraction. 11. What is the decimal equivalent to -J^ ? 12. What are the common fractions equivalent to .375, .( 25, and .875 ? 13. What are the decimals equivalent to -J and |- ? 14. What are the decimals equivalent to ^ and | ? 15. Reduce .33J and .66 to common fractions. 16. Reduce 5 inches to the decimal of a foot. 17. Reduce 5 ounces to the decimal of a pound avoirdu- pois. 18. How many ounces in .583 of a pound Troy ? 19. Reduce 7 grains to the decimal of a pennyweight. 20. How many furlongs in .625 of a iriile ? IX. PROPORTION. 3<64.. Illustration of Ratios. When two quantities of the same kind are compared with regard to their relative value, one of them will be less than, equal to, or greater than, the other ; and will contain the other less than once, exactly once, or more than once. The Ratio of one quantity to another of the same kind, is the quotient resulting from the division of the latter by the former ; the division being expressed in a fractional form, or, more frequently, with the dividend following the divisor with this sign ( ! ) between. : is the sign for the ratio of two quantities. It indicates the ratio of the antecedent, or the quantity preceding the sign, to the consequent, or the quantity which follows the sign. The ratio of two numbers shows what part the dividend is of the divisor. Thus, in comparing 7 dollars with 12 dollars, 4 fathoms with 8 yards, and 11 with 3, we find that 7 dollars is -j^ of 12 dollars, and contains 12 dollars -^ of one. time, 15 170 AKITHMETIC. and that their ratio is 12 : 7 ; that 4 fathoms, being equal to 8 yards, is f of 8 yards, and contains 8 yards f of one time, or exactly once, and that their ratio is 8 : 8, which is called the ratio of equality^ since the two terms of the ratio are equal ; and, finally, that 11 is -V- of 3, and contains 3 -y- of one time, or 3|^times, and that their ratio is 3 : 11. Hence, (87^) to ask what part of 12 dollars is 7 dollars, is the same as to ask what is the ratio of 12 dollars to 7 dol- lars, or of 12 to 7, since -^^ is the part of 12 that 7 is, and also the ratio of 12 to 7. Consequently, any fraction is the ratio of its denominator to its numerator ; and in writing a ratio fractionally, the first number is made the denominator, or divisor, and the second the numerator, or dividend. Thus, 12 : 7 is read, the ratio of 12 to 7, and is the same as -pV* SOil. Exercises in finding the Ratios of Numbers. In like manner^ solve and explain the following prohlems. 1. What part of 7 is 3, and what is the ratio of 7 to 3 ? 2. What part of 5 is 12, (ST.) and what is the ratio of 5 to 12 ? 3. V/hat part of 8 is f, (ll^,) and what is the ratio of 8 tot? 4. What part of 10 is 31, (US,) and what is 10 : 3J ? 5. What part of | is 4, (ISS,) and what is | : 4 ? 6. What part of llf is 5, {\^%) and what is 11| : 5 ? 7. What part of \ is f , (IS^^) and what is |- : -^ ? 8. What part of 12i is 6i, (ISS,) and what is 12^ : 6J ? 9. What part of .1875 is .125, (180.) and what is .1875 :.125? 10. What part of 6.25 is .625, (188.) and what is 6.25 : .625 ? 11. Wliat part of 1.16 is .83, (184,) and what is 1.16 : .83? 12. What part of 12 hours is 5h. 15 m., (228.) and what is 12 h. : 5h. 15m.? 13. What part of 1 10s. is 13s. 6d., (230.) and what is 1 10s. : 13s. 6d. ? 14. What part of 16s. 6d. is $2.50, (254.) and what is 16s. 6d. : $2.50 ? PROPORTION. 171 S{66 Model of a Recitation. 1. Multiply 25 by the ratio of 7 to 3. The ratio of 7 to 3 being ^, to 2^==J^^=10f multiply 25 by the ratio of 7 to 3 is the same as to multiply it by ^, that is, (I445) to take ^ of 25, making 10^, which is the answer required. 2. Multiply 8/^ by 63 : 40. 8yV = -VV-' and 63 : 40 }^dt ^iTi == If J therefore, multiply I 3g X 40 __ Z^ __ 5 5 the denominator by 63, to /0 X $ii 14 ^^ obtain ^, {II45) and mul- 2 7 tiply the numerator by 40, to obtain f . But both terms cf this fraction having the common factors 9 and 8, reduce tie fraction to its lowest terms, (121j) before performing tie operations indicated by the signs, (2i53.) 3. If a man travel 30 miles in 7 hours, what distance \/ould he travel in 12 hours ? If in 7 hours he travel 2oxj^^^6(i = 513 jniles ^^ "^^^^^' ^^ ^ *^^^^ ^ ' ^ ^ * would travel ^ of 30 miles, or ^ miles, (845) and in 12 hours he would travel 12 times as far, or s^^ ^ ^ = ^^ r= 51^ miles. A shorter explanation. 12 hours being J/- of 7 hours, he would travel in 12 hours ^^ of the distance that he would ii. 7 hours. ^ of 30 miles is 3qXi8 aso _ 513 j^jies, (148.) 4. If 5 tons of hay keep 60 sheep through the winter, how much would keep 75 sheep the same time ? 75 sheep being |^ of 60 sheep, -^ = ^ == 6| tons. they would require |4 or J, as much hay. |- of 5 tons is -^^ = ^5- = 6 J tons. S67. Exercises in multiplying by Ratios. In like manner^ solve and explain the following prolilems. 1. If a piece of linen cost $24, what would J of a piece cost ? 2. If 3 chaldrons of coal cost $36, what part of $36, and how much, would 1 chaldron cost ? 172 ARITHMETIC. 3. At $4.20 per box of lemons, what part of $4.20, and how much, would | of a box cost ? 4. At $7.50 per cord, what part of $7.50, and how much, would I of a cord of wood cost ? 5. At $.75 per bushel, what part of $.75, and how much, would 4|- bushels of corn cost ? 6. If 6 horses eat 18 bushels of oats in a week, what part of 18 bushels, and how much, would 5 horses eat ? 7. If 25 lbs. of sugar cost $2.25, what would be the cost of 60 lbs. ? 8. If 5 tons of hay cost $87.50, what part of $87.50, and how much, would 12 tons cost ? 9. At $54 for 9 barrels of flour, what part of 9 barrels, and how much, could be purchased for $186 ? 10. If a vessel sail 480 miles in 5 days, how long would it take her to sail 3000 miles ? 11. If 30 cords of wood cost $200, what part of $200, and how much, would 75 cords cost ? 12. If 3 books cost f of a dollar, what part of J of a dollar and how much, would 8 books cost ? S^8* Reduction of Complex Fkactions. A complex fraction is a fraction in which either term, or both terms are fractions, or mixed numbers. It may be reduced to a simple fraction by multiplying both terms (131) 284 ~%^ by the denominators of the terms. Thus, ~I, or =, are complex fractions, and by multiplying both terms by 5 and 8, or 40, we have V^^'- = VA^- If the denominators of the terms of a complex fraction have a common multiple (131) less than their product, multiply I both terms by that least common multiple. Thus, in =, by "^ . multiplying both terms by 24, we have the simple fractioi? S69. Model of a Recitation. 16. If SJlbs. of butter cost $1-^, what would be the price of 28|lbs. at that rate ? PROPORTION. 173 ^ yw Since $l^,or$|^ y^ v^cvx ^A^ c,A ^s ^^^ P^ice of 8i lbs. rt ./ -^ = $-= $4.80. ^ -V- lbs., divide it by ""XXX^^X^ ""d 65 for the price of Jib. $ and multiply that quo- tient, $3-^-5^-5-, by 8 f )r the price of fib. or 1 lb. (156) I then, since the price of S8f lb. or ^^ lb. is required, divide the price of 1 lb. SyVxVb* ly 5 for the price of ^ lb. and multiply that quotient, ^'tt^fV'x5' by 143 for the price of -if^ lb. or 28f lb. (I495) naking $-Y^|^*f== $2^4 = $4.80, which is the answer r squired. In reducing this fraction, -r^-i^^f-i to its lowest terms, (131) we divide both terms by 11 by canceling the 11 in the cenominator by the 11 which is a factor of 143 in the nume- rator ; 13, the other factor of 143, we cancel by the 13 which is a factor of 65 in the denominator; and 5, the other factor cf 65, we cancel by. the 5 which is a factor of 15 in the r umerator, giving -2^, or ^. Note. In canceling equal factors, there will be less liability t3 mistake, and greater facility in reviewing the process, if one continued line be drawn through the two numbers con- taining the factor to be canceled. A shorter explanation. If Sl^ or, S^-, the price of 8J lb. or V lb. be divided by -V-, (156,) the quotient, TV>f^' will be the price of 1 lb. ; and if this price of 1 lb. be multiplied by 283-, or -i^, the number of pounds whose price is required, (149,) the product $\^-^^V' = HS = $4.80, must be the answer required. 285. ^^ Or shorter still. 28f lb. being -^f , or =^ of 8| lb. or ^ lb. ^ JA3 143 would cost =: of the price of -^^ lb. ==: of $l^y, or $4t' ^^ ^\'^i^kh = $' = $4.80, as before. 370* Observation. Observe, that the process^ by either explanation, (Q69^) is the same, and consists of multiplying^ the given price of a given quantity by the ratio of the given quantity to the REQUIRED quantity, or by the part of the given quantity that 15=^ 174 ARITHMETIC. the required quantity must he. Labor often may he saved by mentally reducing the ratio to simpler terms ^ (121 5) hefore writing it, 271, Exercises in Multiplying by Complex Ratios. In like manner, solve and explain the following problems. 1. If 37 yards of broad cloth cost $185, what would 5f yards cost ? 2. If If of a cask of wine cost $12.50, what would 6 such casks cost ? 3. At $3f for 4f yards of satinet, what would be the cost of 25 yards ? 4. If 12 days' work cost $16|, what would 5f days' work cost ? 5. If 1^ of a bushel of corn cost $|-, what is that a bushel ? 6. If IJ barrels of flour serve a family 1^ weeks, how long would 7^ barrels serve them ? 7. If a (Company of workmen mow 72^ acres in 12 1 days, how many acres would they mow, at the same rate, in Sf days ? 8. If 3 yds. 3 qrs. of cassimere cost $10, what would 5 yards cost ? 9. If 18 gals. 3 qts. of wine cost $33.75, what would 43 gals. 3 qts. cost ? 10. If 2 roods, 25 rods of land cost $42, what would 5 acres, 3 roods cost, at that rate ? 11. If ^jj sterling money make $2, how much sterling money is equal to $12J ? 272. Illustration of the Inverse Ratio. If 5 men could build a wall in 32 days, in how long time could 9 men build it ? It would take 1 man 5 times as long as it would 5 men, or 5 times 32 days ; but it would take 9 men only J as long as it would 1 man,, or ^ of 5 times 32 days, which is | of 32 days, or ^^l^=:i|.o = 17 J- days, the answer required. S73. Observation. Observe, (272^) that, though 9 men arc % of 5 men, it would not take them f of the time that it would take 5 men to build the wall, but rather ^ |- of that time ; but f w f PROPORTION. 176 i?JVERTED. Hence, f in this example, and the ratio corre- svonding to it in similar cases, being called a direct ratio, f and the ratio corresponding to it in similar cases, is called en INVERSE RATIO. tl74:m Model of a Recitation. If 12 cows consume a quantity of hay in 90 days, how inany cows would consume the same hay in 30 days ? To consume the same hay in 30 days would require f, or (131) 3 times as many cows as would consume it in 90 (lays. 3 times 12 cows are 36 cows, the answer required. ^27^. Exercises in Multiplying by Inverse Ratios. In like manner, solve and explain the following problems. 1. If 9 horses consume a ton of hay in 32 days, how long .vould it take 12 horses to consume the same hay ? 2. If 72 men could do a job of work in 15 months, how inany men, working*at the same rate, would do the same job ] n 2 years ? 3. If a barrel of flour last a family of 6 persons 12 weeks, Jiow long would a barrel last them if the family be increased i;o 8 persons ? 4. If a pail holding 10 quarts be emptied 200 times to fill a cistern, how much would that vessel hold which must be emptied 75 times to fill the same cistern ? 5. 4s. 6d. sterling money being equal to 5s. Canada cur- rency, how much sterling money would cancel a debt of 18 in Quebec ? 6. How much Canada currency would cancel a debt of 36 in London ? 7. 5s. Canada currency being equal to 6s. N. E. currency, how much Canada currency is equal to 45 N. E. currency? 8. How much sterling money is equal to 36 N. E. currency ? 9. 6s. N. E. currency being equal to 8s. N. Y. currency, how many N. E. pounds are equal to 72 N. Y. pounds ? 10. A piece of land 8 rods wide and 20 rods long is an acre ; then how long must that acre be which is 12 rods wide ? 11. A board 9 in. wide and 16 in^long being a square foot, how wide must that board be which contains 1 sq. foot, and is 16 feet long ? it * 176 ARITHMETIC. 12. If a Stick of timber, the end of which contains 216 sq. inches, must be 37J feet long to be a ton, how long must a stick be to measure a ton, the end of which contains 288 sq. inches ? 13. If the contents of a cylindrical tube, which measures 18 inches in length and 144 sq. inches on one end, be emptied into another tube the end of which should measure 16 sq. inches, how high would the water rise ? 14. How many yards of cloth | of a yard wide would be equal to 12| yards 1| yards wide ? 15. How many yards of cloth, 1| yds. wide, would be equal to 91^ yds. f of a yard wide ? 16. What quantity of wheat, at $1.25 a bushel, should be given for 10 barrels of flour, at $5.25 a barrel ? 17. What quantity of sugar, at SJcts. a pound, would pay for board 12 weeks, at $2.75 a week ? 18. In how many weeks, at SIOOO per annum, could a man earn as much as another man could in 13 weeks, at $700 per annum ? 376* Illustration of the Principles of Proportion. A large and a sihall map of the U. S. in order to be correct representations of the country, must be of the same shape ; and the states, mountains, lakes, rivers, cities, towns, &;c.j must have the same relative distaiices on each map ; that is, all the distances on each map, must be in 'proportion to the corresponding distances on the other map. Thus ; if New York is \ as far from Washington as Boston is on one map, it must also be \ as far from W. as B. is on the other map. If then, on the larger map, the two distances of B. and N. Y. from W. be 12 inches, and \ of 12 iiiches, or 6 inches, and on the smaller map, the distance of B. from W. be 4 in., the distance of N. Y. from W. must he \ of ^ inches, or 2 in. That is, the ratio (264) of the two distances of B. and N. Y. from W. on one map, must be equal to the ratio of the cor- responding distances on the other map. Thus, 12 : 6 == 4 : 2. This expression constitutes what is called a pro- vortion. Observe, then, that a proportion is composed of two equal ratios, and that a ratio is the relation of two quantities of the same kind, in regard to what part (87) of the first, the second is, or how many times the first is contained by the PROPORTION. 177 ee'.ond Thus, in the proportion 12 6 = 4:2, the first raio 12 : 6, or -fir, is equal to the second ratio 4 : 2, or |, since each is equal to ^. This proportion is read: The ra Jo of 12 to 6 equals the ratio of 4 to 2 ; or 12 is to 6 as 4 is to 2 ; or, as 12 is to 6 so is 4 to 2. The four quantities forming a proportion are called pro- portionals, or, the terms of the pi'oportion. The first and fourth terms of a proportion are called the ex '^.rentes, and the second and third, the means. Also, the first terms, or divisors in ratios, are called the atitecedentSj and the second terms, or dividends, are called the CO tsequents. Two equal fractions may become a proportion, by placing th3 denominators for antecedents, and the numerators for consequents. And, any four numbers, arranged like propor- tiimals, form a correct proportion, if the product of the means he equal to the product of the extremes, since these products, in a correct proportion, will always be equal ; for, in a- correct proportion, the two ratios, or fractions 12 : 6==4 : 2 being equal, if they be reduced to a com- -5-^2- = f mon denominator, ( 139,) by multiply- y6^^= |gJ-| ing both terms of each by the denomina- || = || tor of the other, the numerators will he equal also. But one of these numerators is the product of the means, and the other is the product of the extremes. This truth is of great practical utility in the solution of problems which involve proportion ; since, by its application, any three terms of a proportion are sufficient for ascertaining the remaining term. For, if the term wanting be an extreme, it may be ascertained by dividing the product of the means, (which is also the product of the extremes, one of which is known,) by the known extreme, (117) ; or, if the term wanting be a mean, it may be ascertained by dividing the product of the extremes, (which is also the product of the means, one of which is known,) by the known mean. Thus, in the proportion, 12 : 6== 4 : 2, the product of the means divided by the^^r^^ extreme, is ^-^^ = 2, the second extreme ; or divided by the second extreme, is ^^= 12, the first extreme ; and the product of the extremes^ divided by the first mean, is ^^^=4, the second mean; or, divided by the second mean, is ^^^ = 6, the first mean.