J9ublfsI)tU for t|)e ?pnr of SWarren Colimrn, bi> HURD AND HOUGHTON, 13 Astor Place, New York. LEE AND SHEPARD, 149 Washington" St., Boston. Entered aee*rdinf x Act rChrrM in th year 1863. i.y Trmperanc* C. Colhum, Wid'>w of Wnrren Colhurn, in the Clerk'* Olfice of jji Dw- tri^t Curt of the Distnc- of .M ,.^chu.ctu. ; : l II Sc The The Young Pioneers 1 25 By Oliver Optic. Outward Bound; or, Young Amer- ica Afloat 150 Young Hunters' Library. The Australian Wanderers 1 50 The 'African Crusoes 1 oO Anecdotes of Animals -1 50 Anecdotes of Birds 1 50 One Hundred New Illustrated Juvenile Books. The Jutland Series. The Sand Hills of Jutland $1 50 Yarns of an Old Mariner 1 50 Schoolboy Days 1 60 Great Men and Gallant Deeds 1 50 By James De Millo. TheB.O. W. C 160 The Boys of Grand Pr6 School 1 50 Lost hi the Fog 1 50 By May Manner ing. Climbing the Rope 1 00 Billy Grimes's Favorite 1 00 The Cruise of the Dashaway 1 00 The Little Spaniard 1 00 Salt Water Dick 1 00 Little Maid of Oxbow 1 00 The Proverb Stories. Birds of a Feather 1 00 Fine Feathers do not make Fine Birds 100 Handsome Is that Handsome Does 1 00 A Wrong Confessed is Half Ke- dressed 1 00 Actions Speak Louder than Words 1 00 One Good Turn deserves Another 1 00 By Sophie May. Little Prudy 75 Little Prudy 's Sister Susy 75 Little Prudy's Captain Horace. ... 76 Little Prudy 's Cousin Grace. ..... 75 Little Prudy 's Story-book 75 Little Prudy 'B Dotty Dimple 75 Dotty Dimple at Her Grandmoth- er's 75 Dotty Dimple at Home 75 Dotty Dimple Out West 75 Dotty Dimple at Play 75 Dotty Dimple at School 75 Dotty Dimple's Flyaway 75 Little Folks Astray 75 Prudy Keeping House 75 By Miss Louise M. Thnrston. How Charley Roberta became a Man $100 How Eva Roberta gained her Edu- cation 100 Charley and Eva's Home in the West 100 Spriiigdale Stories. Adele 76 Eric 75 Herbert 75 Nettie's Trial 76 Johnstone's Farm 76 Ennisfellen 76 Wonderland Stories. Alice's Adventures in Wonder- land 160 Through the Looking-glass, and what Alice saw there 1 60 The House on Wheels: or, The Stolen Child .1 50 Letters Everywhere ; or, Rhymes and Stories for Children . .1 60 By George M. Baker. Amateur Dramas ..160 ' The Mimic Stage ; .1 60 The Social Stage 1 60 By Miss A. M. Douglas. Kathie's Three Wishes 1 00 Kathie's Aunt Ruth 1 00 Kathie's Summer at Cedarwood.-l 00 Kathie's Soldiers 1 00 By William Everett. Changing Base 125 Double Play 1 25 Instructive and Entertain- ing. Nat, the Navigator 1 00 Ned Nevins, the Newsboy 1 50 Dialogues from Dickens 1 50 Send for our Complete Descriptive Catalogue , mailed free. LEE & SHEPARD, Boston; LEE, SHEPARD, & DILLINGHAM, New York. ffARREN COLBUEN'S FmST INTELLECTUAL ARITHMETIC, INDUCTIVE METHOD INSTRUCTION. BY WARREN COLBURN, A. M. WITH AN INTRODUCTION TO WRITTEN ARITHMETIC BY HIS SON, WARREN COLBURN, AND AN INTRODUCTION BY GEORGE B. EMERSON. PUBLISHED FOR THE HEIRS OF. WARREN COLBURN BY HURD AND HOUGHTON, 13 ASTOR PLACE,, NEW YORK H. H. BANCROFT & CO., SAN FRANCISCO, CAL. OA JEntered according to Act of Congress, in the ye) TKMPBBANCB 0. COLBUBN, Widow of Warren Colburn, ID the Clerk's Office of the District Court of the District of Masaachusetti Entered according to Act of Congress, in the year 1868, by TIMPERANCH C. COLBUBN, Widow of Warren Golburn^L to the Clerk's Office of the District Court of the District of Massach Entered according to Act of Congress in the year 1863, by TJOIPERANCJ C. COLBURN, Widow of Warren Colburn, In the Clerk's Office of the District Court of the District of Massachusetts APVERTISEMENT. fcDUcXrjoi DI THIS book has returned by purchase into the pos session of the heirs of the late WARREN COLBURN, and will hereafter be published for their benefit. The Introduction, by GEORGE B. EMERSON, gives some valuable hints as to the proper use of the book. The Original Preface, in which the design and plan of the work are stated at length by the Author, has been restored. A careful perusal of this Introduction and Preface is earnestly recommended to all teachers who make use of the book. H. 0. HOUGHTON. Uambridye, June, 1863. INTRODUCTION. THIS little book has now been in constant use foi forty years, and its value has been proved by the entire and admiring approbation of thousands of the best teachers. Yet its very simplicity has prevented mauy persons from seeing how really profound and comprehen- sive it is, and that it actually develops every essential prin- ciple in elementary arithmetic. Want of attention and the previous use of poorer Books have also misled many persons as to the proper and true mode of using it. It is strictly a mental arithmetic, and, if faithfully used in the way intended by the author, it evolves from the mind of the learner himself, in a perfectly easy and natural manner, a knowledge of the principles of arithmetic, and the power of solving, mentally and almost instantly, every question likely to occur in the every-day business of com- mon life. It can be well taught only by a teacher who perfectly understands it, and who knows how to teach. Such a teacher will not allow the lesson to be previously studied by the pupil. Each section is intended to teach some one process up to a certain point. If, in the course of the sec- tion, questions occur which the class cannot readily solve without previous study, the teacher has only to interpose, at the point where the class fails, or begins to fail, addi- 733 4 INTRODUCTION. "tibri&l q&eatioW b'f 'the same kind, somewhat easier than those in the book. If, at the end of the section, the class be not perfectly ready in the solution of the questions, the teacher ought to go over the section again, with the class, or to add, at the end of the section, a sufficient number of similar questions to render the solution easy and instanta- neous. By allowing the class to study the lesson beforehand, not only is much time lost, but the exercise is turned into a poor sort of mechanical process not much better than the common ciphering. Its mental character ceases almost en- tirely. As an exercise in exact reasoning, the mental solution of questions is very valuable ; not less so than geometrical demonstration, and preferable in the fact that it is purely mental. It is thus an admirable preparation for the attain- ment of the faculty of ready and sharp logic, to be applied to the business of common life, where the mental habit alone is available. Its great value for children is in its shortness and its clearness. For example : What is J. ot | of | of | ? Answer : of o f of is ; for of * A? 4 of is I ; i of t * 1 5 therefore,*, of of 3 of | is L. Similar examples are found jn almost every section in the book. Now, the going faithfully through many such processes, every day for years, at the age at which mental habits are formed, must have the effect of forming habits of clearness, accuracy, and exact reasoning, which, once fully formed, will always exert a happy influence over the mind. This book ought not to be used with beginners only. Its effect upon those far advanced is even better and more striking. It might be continued, with the greatest advan- INTRODUCTION/' 5 tage, till the end of every course of* MlstVuciionJ and*, 'ft* sc' ' continued, would Lave the constant effect of sharpening the perception and varying the ability of analysis, of giving perfect readiness in the mental solution of all com- mon problems, and of fixing the habit of clearness, quick- ness, and accuracy in the process of exact reasoning. GEORGE B. EMERSON. May, 1863. THE ORIGINAL PREFACE. As soon as a child begins to use his senses, nature con- tinually presents to his eyes a variety of objects ; and one of the first properties which he discovers is the relation of number He intuitively fixes upon unity as a measure and from this he forms the idea of more and less ; which is the idea of quantity. The names of a few of the first numbers are usually learned very early ; and children frequently learn to count as far as a hundred before they learn their letters. As soon as children have the idea of more and less, and the names of a few of the first numbers, they are able to make small calculations. And this we see them do every day about their playthings, and about the little affairs which they are called upon to attend to. The idea of more and less implies addition ; hence they will often perform these operations without any previous instruction. If, for example, one child has three apples, and another five, they will readily tell how many they both have ; and how many one has more than the other. If a child be requested to bring three apples for each person in the room, he will cal- culate very readily how many to bring, if the number does not exceed those he has learnt. Again, if a child be re- quested to divide a number of apples among a certain number of persons, he will contrive a way to do it, and will ORIGINAL PREFACE. Lave. The method which chil- dren take to do these things, though always correct, it not always the most expeditious. The fondness which children usually manifest for thew exercises, and the facility with which they perform them, seem to indicate that the science of numbers, to a certain extent, should be among the first lessons taught to them. To succeed in this, however, it is necessary rather to furnish occasions for them to exercise their own skill in performing examples, than to give them rules. Thoy should oe allowed to pursue their own method first, and then "they should be made to observe and explain it ; r.nd if it was not the best, some improvement should !> Mi;_in\ 4, &c., is one half, one third, one fourth, &c. ; and also knowing J, | ? 4, &c. of a number, to find that number. . These combinations contain all the most common and most useful operations of vulgar fractions. But being ap- plied only to numbers which are exactly divisible into these fractional parts, the pupil will observe no principles but multiplication and division, unless he is told of it. In fact, fractions contain no other principle. The examples are so arranged that almost any child of six or seven years old will readily comprehend them. And the questions are asked in such a manner that, if the instructor pursues the method explained in the Key, it will be almost impossible for the pupil to perform any example without understand- ing the reason of it. Indeed, in every example which he performs, he is obliged to go through a complete demonstration of the principle by which he does it ; and at the same time he does it in the simplest way possible. These observations apply to the remaining part of the book. These principles are sufficient to enable the pupil to perform almost all kinds of examples that ever occur. He will not, however, be able to solve questions in which it is necessary to take fractional parts of unity, though the principles are the same. After section sixth, there is a collection of miscellaneous s, in which are contained almost all the kinds that : : '*Tk ORIGINAL PREFACE. are none, however, which the prin ciples explained are not sufficient to solve. In section eighth and the following, fractions of unity are explained, and, it is believed, so simply as to be intelligible to most pupils of seven or eight years of age. The opera- tions do not differ materially from those in the preceding sections. There are some operations, however, peculiar to fractions. When the pupil is made familiar with all the principles contained in this book, he will be able to perform all exam- ples, in which the numbers are so small that the opera- tions may be 'performed in the mind. Afterwards he has only to learn the application of figures to these operations, and his knowledge of arithmetic will be complete. The following are some of the principal difficulties which a child has to encounter in learning arithmetic, in the usual way, and which are seldom overcome. First, the ex- amples are so large that the pupil can form no conception of the numbers themselves ; therefore it is impossible for him to comprehend the reasoning upon them. Secondly, the first examples are usually abstract numbers. Thi? increases the difficulty very much ; for even if the numbers were so small that the pupil could comprehend them, he would discover but very little connection between them and practical examples. Abstract numbers, and the op- erations upon them, must be learned from practical exam- ples ; there is no such thing as deriving practical examples from those which are abstract, unless the abstract have been first derived from those which are practical. Thirdly, the numbers are expressed by figures, which, if they were used only as a contracted way of writing numbers, would be much more difficult to be understood at first than the THE ORIGINAL PREFACE 13 numbers written at length in words. ^ .But 1 4;^y ^ are 'nbt used merely as words, they require operations peculiar to themselves. They are, in fact, a new language, which the pupil has to learn. The pupil, therefore, when he com- mences arithmetic, is presented with a set of abstract num- bers, written \f\\hfgures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures sig- nify, and 'what is meant by addition, subtraction, multiplica- tion, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is, how to per- form these operations on figures. He can perhaps translate the figures into words, but this is useless, since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has not the least conception. After the abstract examples, a few practical examples are usually given, but these again are so large that the pupil cannot reason upon them, and consequently he could not tell whether he must add, subtract, multiply, or divide, even if he had an adequate idea of what these operations are. The common method, therefore, entirely reverses the natural process ; for the pupil is expected to learn general principles, before he has obtained the particular ideas of which they are composed. The usual mode of proceeding is as follows. The pupil learns a rule, which, to the man that made it, was a gen- eral principle ; but with respect to him, and oftentimes to the instructor himself, it is so far from it, that it hardly de- serves to be calkd even a mechanical principle. He per* 14 ' THE ORIGINAL PKEFACF fdrftis ^tAe- exzftipkA, and makes the answers agree with those in the book, and so presumes they are right. He la soon able to do this with considerable facility, and is then supposed to be master of the. rule. He is next to apply his rule to practical examples ; but if he did not find the examples under the rule, he would never so much as mis- trust they belonged to it. But finding them there, he applies his rule to them, and obtains the answers, which are in the book, and this satisfies him that they are right. In this manner he proceeds from rule to rule through the book. When an example is proposed to him, which is not in the book, his sagacity is exercised, not in discovering the operations necessary to solve it, but in comparing it with the examples which he has performed before, and endeav- oring to discover some analogy between it and them, either in the sound, or in something else. If he is fortu- nate enough to discover any such analogy, he finds what rule to apply ; and if he has not been deceived in tracing the analogy, he will probably solve the question. His knowledge of the principles of his rules is so imperfect, that he would never discover to which of them the example belongs, if he did not trace it, by some analogy, to the examples which he had found under it. These observations do not apply equally to all ; for some will find the right course themselves, whatever obstacles be thrown in their way. But they apply to the greater part ; and it is probable that there are very few who have not experienced more or less inconvenience from this mode of proceeding. Almost all, who have ever fully understood arithmetic, have been obliged to learn it over again in their own way. And it is not too bold an assertion to say, that THE ORIGINAL PREFACE. 15 no man ever actually learned mathematics in any other method than by analytic induction ; that is, by teaming the principles by the examples he performs ; and not by learning principles first, and then discovering by them how the examples are to be performed. THE BOY WITHOUT A GENIUS. MR. WISEMAN, the schoolmaster, at the end of his summer vacation, received a new scholar, with the following letter : SIR, This will be delivered to you by my son, Samuel, whom I beg leave to commit to your care, hoping that by your well-known skill and attention you will be able to make something of him ; which, I am sorry to say, none of his masters have hitherto done. He is now eleven, and yet can do nothing but read his mother-tongue, and that but indiffer- ently. We sent him at seven to a grammar school in our neighborhood ; but his master soon found that his genius was not turned to learning languages. He was then put to writ- ing, but he set about it so awkwardly that he made nothing of it. He was tried at accounts, but it appeared that he had no genius for that neither. He cpuld do nothing in geography for want of memory. In short, if he has any genius at all, it does not yet show itself. But I trust to your experience in cases of this nature to discover what he is fit for, and to instruct him accordingly. I beg to be favored shortly with your opinion about him, and remain, sir, Your most obedient servant, HUMPHREY ACRES. When Mr. Wiseman had read this letter, he shook his head, and said to his assistant, A pretty "subject they have sent us here ! a lad that has a great genius for nothing at all. But perhaps my friend Mr. Acres expects that a boy should show a genius for a thing before he knows anything about it, no uncommon error! Let us see, however, what the youth looks like. I suppose he is a human creature at least. Master Samuel Acres was now called in. He came, hang- ing down his head, and looking as if he was going to be flogged. 16 THE ORIGINAL PREFACE. Come hither, my dear ! said Mr. Wiseman. Stand by me, and do not be afraid. Nobody will hurt you. How old are you? Eleven last May, sir. A well-grown boy of your age, indeed. You love play, I dare say ? Yes, sir. What, are you a good hand at marbles 1 Pretty good, sir. And can spin a top and drive a hoop, I suppose ? Yes, sir. Then you have the full use of your hands and fingers f Yes, sir. Can you write, Samuel ? I learned it a little, sir, but I left it off again. And why so ? Because I could not make the letters. No ! why, how do you think other boys do ? Have they more fingers than you ? No, sir. Are you not able to hold a pen as well as a marble ? Samuel was silent. Let me look at your hand. I see nothing here to hinder you from writing as well as any boy in the school. You can read, I suppose ? Yes, sir. Tell me, then, what is written over the school-room door ? Samuel, with some hesitation, read, WHATEVER MAN HAS DONE MAN MAY DO. Pray, how did you learn to read ? Was it not with taking pains ? Yes, sir. Well, taking more pains will enable you to read better. Do you know anything of the Latin grammar ? No, sir. Have you never learned it ? I tried, sir, but I could not get it by heart. Why, you can say some things by heart. I dare say you can tell me the names of the days of the week in their order Yes, sir, I know them. And the months in the year, perhaps ? Yes, sir. And you could probably repeat the names of your brother? and sisters, and all your father's servants, and half the people in the village besides ? I believe I qould, sir THE ORIGINAL PREFACE. 17 Well, and is hie, hcec, hoc, more difficult to remember than these ? Samuel was silent. Have you Learned a. ything of accounts ? I went into addition sir, but I did not go on with it. Why so ? I could not do it, sir. How many marbles can you buy for a penny ? Twelve new ones, sir. And how many for a half-penny ? Six. And how many for two-pence ? Twenty-four. If you 'were to have a penny a day, what would that make in a week ? Seven-pence. But if you paid two-pence out of that, what would you have left ? Samuel studied a while, and then said, five-pence. Right. Why, here you have been practising the four great rules of arithmetic, addition, subtraction, multiplication, and division. Learning accounts is no more than this. Well, Samuel, I see what you are fit for. I shall set you about nothing but what you are able to do ; but observe, you must do it. We have no I can't here. Now, go among your school- fellows. Samuel went away, glad that his examination was over, and with more confidence in his powers than he had felt before. The next day he began business. A boy less than himself was called out to set him a copy of letters, and another was appointed to hear him in grammar. He read a few sentences hi English, that he could perfectly understand, to the master himself. Thus by going on steadily and slowly, he made a sensible progress. He had already joined his letters, got all the declensions perfectly, and half the multiplication-table, when Mr. Wiseman thought it time to answer his father's letter; whicfi he did as follows : SIR, I now think it right to give you some information Concerning your son. You perhaps expected it sooner, but [ always wish to avoid hasty judgments. You mentioned in your letter that it had not yet been discovered which way his genius pointed. If by genius you meant such a decided bent of mind tc any one pursuit as will lead to excel with little or no labor or instruction, I must say that I have not met with 18 THE ORIGINAL PREFACE such a quality in more than three or four boys in my life, and your son is certainly not among the number. But if you mean only the ability to do some of those things which the greater part of mankind can do when properly taught, I can affirm that I find in him no peculiar dt riciency. And whether you choose to bring him up to trade < r to some practical pro- fession, I see no reason to doubt that he may in time become sufficiently qualified for it. It is my favorite maxim, sir, that everything most valuable in this life may generally be acquired by taking pains for it. Your son has already lost much time in the fruitless expectation of finding out what he would take up of his own accord. Believe me, sir, few boys will take up anything of their own accord but a top or a marble. I will take care, while he is with me, that he loses no more time this way, but is employed about tilings that are fit for him, not doubting that we shall find him fit for them. I am, sir, yours, &c., SOLON WISEMAN. Though the doctrine of this letter did not perfectly agree frith Mr. Acres' notions, yet being convinced that Mr. Wise- man was more likely to make something of his son than any of his former preceptors, he continued him at his school for some years, and had the satisfaction to find him going on in a steady course of gradual improvement. In due time a pro- fession was chosen for him, which seemed to suit his temper and talents, but for which Re had no particular turn, having never thought at all about it. He made a respectable figure in it, and went through the world with credit and usefulness, though without a genius. Mrs. Barbauld. ARITHMETIC. PART I. SECTION I. A.* 1. How many thumbs have you on your right hand? how many on your left? how many on both together ? 2. How many hands have you ? 3. If you have two nuts in one hand, and one in the other, how many have you in both ? 4. How many fingers have you on one hand ? 5. If you count the thumb with the fingers, how many will it make ? 6. If you shut your thumb and one finger, and leave the rest open, how many will be open ? 7. If you have two cents in one hand, and two in the other, how many have you in both ? 8. James has two apples, and William has three ; if James gives his apples to William, how many will William have? 9. If you count all the fingers on one hand, and two on the other, how many will there be ? 10. George has three cents, and Joseph has four; liow many have they both together ? * The first questions in this section are intended for very young children. It will be well for the instructor to giv i a great many more of this kind. Older pupils may omit these. 20 AKITHMETIC. [Part 1 11. Robert gave five cents for an orange, and two for an apple ; how many did he give for both ? 12. If a custard cost six cents, and an apple two cents, how many cents will it take to buy an apple and a custard ? 13. If you buy a pint of nuts for five cents, and an orange for three cents, how many cents would you give for both ? hu\v many more for the nuts than for the orange? 14. If an ounce of figs is worth six cents, and a half a pint of cherries is worth three cents, how much are they both worth ? 15. Dick had five plums, and John gave him four more; how many had he then? 1 6. How many fingers have you on both hands ? 17. How many fingers and thumbs have you on both hands? 18. If you had six marbles in one hand, and four in the other, how many would you have in the one more than in the other? how many would you have in both hands ? 19. David had seven nuts, and gave three of them to George; how many had he left? 20. Two boys, James and Robert, bought seven marbles apiece ; James gave Robert four of Ids ; how many had each then ? 21. A boy, having eleven nuts, gave away three of them ; how many had he left ? 22. If you had eight cents, and your papa should give ypu five more, how many would you have ? 23. A man bought a sheep for eight dollars, and a calf for seven dollars ; what did he give for both? 24. A man bought a barrel of flour for eight dol- lars, and sold it for four dollars more than he gave foi it ; how much did he sell it for ? 25. A man bought a hundred-weight of su^ar foi Sect. I.] ARITHMETIC 21 nine dollars, and a barrel of flour for seven dollars how much did he give for the whole ? 26. A man bought three barrels of cider for eight dollars, and ten bushels of apples for nine dollars how much did he give for the whole ? 27. A man bought a firkin of butter for twelve dol- lars, but, it being damaged, he sold it again for eight dollars ; how much did he lose ? 28. A man bought three sheep for fifteen dollars, but could not sell them again for so much by eigh* dollars ; how much did he sell them for ? 29. A man bought sixteen pounds of coffee, and lost seven pounds of it as he was carrying it home ; how much had he left? 30. A man bought nineteen pounds of sugar, and having lost a part of it, he found he had nine pounds left ; how much had he lost ? 31. A man, owing fifteen dollars, paid nine dollars of it ; how much did he then owe ? 32. A man, owing seventeen dollars, paid all but seven dollars ; how much did he pay ? B. 1. Two and one are how many ? 2. Two and two are how many ? 3. Three and two are how many ? 4. Four and two are how many ? 5. Five and two are how many ? 6. Six and two are how many ? 7. Seven and two are how many ? 8. Eight and two are how many ? 9. Nine and two are how many ? 10. Ten and two are how many ? 1 1 . Two and three are how many ? 12. Three and three are how many ? 1 3. Four and three are how many ? 14. Five and three are how many ? 15. Six an/, three are how many ? 22 ARITHMETIC. i Azrf I, 16. Seven and three are how many? 17. Kijrht and three are how many? 18. Nine and three are how many? 19. Ten and three are how many? 20. Two and lour are how many? 21. Three and four are how many ? 22. Four and four are how many? 23. Five and four are how many!? 24. Six and four are how many ? 25. Seven and four are how many? 26. Eight and four are how many ? 27. Nine and four are how many ? 28. Ten and four are how many ? 29. Two ajid five are how many ? 30. Three and five are how many ? 31. Four and five are how many? 32. Five and five are how many ? 33. Six and live are how many? 34. Seven and five are how many ? 35. Eijht and live are how many ? 36. Nine and five are how many? 37. Ten and five are how many ? 38. Two and six are how many ? 39. Three and six are how many ? 40. Four and six are how many ? 41. Five and six are how many ? 42. Six and six are how many ? 43. Seven and six are how many ? 44. Eight and six are how many ? 45. Nine and six are how many ? 46. Ten and six are how many ? 47. Two and seven are how many ? 48 Three and seven are how many ? 49. Four and seven are how many ? 50. Five and seven are how many ? 51. Six and seven are how many ? 52. Seven and seven are how many ? Seat. 1.1 ARITHMETIC. 23 53. Eight and seven are how many ? 54. Nine and seven are how many ? 55. Ten and seven are how many ? 56. Two and eight are how many ? 57 Three and eight are how many ? 58. Four and eight are how many ? 59. Five and eight are how many ? 60. Six and eight are how many ? 61. Seven and eight are how many ? 62. Eight and eight are how many ? 63. Nine and eight are how many ? 64. Ten and eight are how many ? 65. Two and nine are how many ? 66. Three and nine are how many ? 67. Four and nine are how many ? 68. Five and nine are how many ? 69. Six and nine are how many ? 70. Seven and nine are how many ? 71. Eight and nine- are how many ? 72. Nine and nine are how many ? 73. Ten and nine are how many ? 74. Two and ten are how many ? 75. Three and ten are how many ? 76. Four and ten are how many ? 77. Five and ten are how many? 78. Six and ten are how many ? 79. Seven and ten are how many ? 80. Eight and ten are how many ? 81. Nine and ten are how many ? 82. Ten and ten are how many ? C. 1 Two and one are how many ? 2 Two and two are how many ? 3. Three and two are how many ? 4. Five and two are how many ? 5. Four and two are how many ? 6. Six and two are how many ? B4 ARITHMETIC. [Part 1 7. Eight and two are how many ? 8. Five and three are how many ? 9. Seven and three are how many? 10. Four ami three are how many? 1 1. Two and three are how many ? 12. Two and six are how many ? 13. Two and eight are how many ? 14. Six and three are how many? 15. Three and four are how many ? 16. Three and six arc how many ? 1 7. Two and seven are how many ? 18. Ten and two are how many? 19. Two and four are how many ? 20. Three and seven are how many ? 21. Four and four are how many ? 22. Five and four are how many ? 23. Seven and two are how many ? 24. Two and five are how many ? 25. Three and three are how many? 26. Four and five are how many ? 27. Nine and two are how many ? 28. Three and five are how many ? 29. Two and ten are how many ? 30. Three and eight are how many ? 31. Ten and three are how many ? 32. Two and nine are how many ? 33. Foui and six are how many ? 34. Eight and three are how many ? 35. Seven and four are how many ? 36. Nine and three are how many ? 37. Six and four are how many ? 38. Five and five are how many ? 39. Three and nine are how many ? 40. Four and seven are how many ? 41. Six and five are how many ? 42. Three and ten are how many ? 43. Eight and four are how many ? Sect. 1.] ARITHMETIC. 25 44. Five and eight are how many ? 45. Four and nine are how many ? 46. Five and six are how many ? 47. Ten and four are how many ? 48. Seven and five are how many ?^ 49. Six and six are how many ? 50. Nine and four are how many ? 51. Eight and five are how many ? 52. Five and nine are how many ? 53. Four and ten are how many ? 54. Six and seven are how many ? 55. Four and eight are how many ? 56. Nine and five are how many ? 57. Six and eight are how many ? 58. Ten and five are how many ? 59. Seven and six are how many ? 60. Eight and seven are how many ? 61. Six and nine are how many ? 62. Seven and seven are how many? 63. Eight and six are how many ? 64. Ten and six are how many ? 65. Eight and eight are how many ? 66. Nine and seven are how many ? 67. Ten and eight are how many ? 68. Six and ten are how many ? 69. Five and seven are how many ? 70. Nine and six are how many ? 71. Seven and eight are how many ? 72. Eight and nine are how many ? 73. Nine and nine are how many ? 74. Five and ten are how many ? 75. Seven and nine are how many ? 76. Nine and eight are how many ? 77. Eight and ten are how many ? 78. Ten and nine are jiow many ? 79. Seven and ten .are how many? 80. Nine and ten are how many ? 81. Ten and ten are how many ? 86 ARITHMETIC. [Par/ 1 D. 1. Three boys, Peter, John, and Oliver, gave some money to a beggar. Peter gave seven cents : John, four cents ; and Oliver, three cents ; how many did they all give him ? 2. How maiiy did Peter give more than Oliver ? 3. Frank had nine pears, and gave three of them to Harry ; how many had he left ; and how many more than Harry had he then? 4. Dick had ten peaches, Harry twelve, and Charles thirteen; Dick gave three to Stephen, Harry gave him six, and Charles gave him five; how many had Stephen ? and how many had each lefl ? 5. A boy had twenty apples, and gave them to his companions, as follows : to one he gave three ; to an- other, two ; to another, four ; and to another, five ; how many did he give away ? and how many had he left? 6. A boy gave to one of his companions eight peaches ; to another, six ; to another, four ; and kept two himself; how many had he at first ? 7. A boy went to the confectioner's and bought three cakes of gingerbread, for which he gave a cent apiece ; two buns, for which he gave three cents apiece ; one custard for four cents, and one orange for six cents ; how many cents did he spend for the whole ? 8. A boy, having twenty-five cents, bought one quart of cherries for eight cents, one orange for six cents, and gave away three cents ; how many cents had he left*? 9. A boy, bought a box for eighteen cents, and gave eight cents to have it painted, and then sold it for thirty-two cents ; how much did he gain by the bar- gain ? 10. A man bought a sleigh for seventeen dollars, and gave nine dollars to have it, repaired and painted, and then sold it for twenty-three dollars ; how much did he lose by the bargain ? Sect. 1.] ARITHMETIC. 27 11. Eleven and two are how many ? 1 2. Eleven and three are how many ? 1 3. Eleven and four are how many ? 1 4. Eleven and five are how many ? 1 5. Eleven and six are how many ? 1 6. Eleven and seven are how many ? 17. Eleven and eight are how many ? 18. Eleven and nine are how many? 19. Eleven and ten are how many? 20. Twelve and two are how many ? 21. Twelve and three are how many ? 22. Twelve and four are how many ? 23. Twelve and five are how many ? 24. Twelve and six are how many ? 25. Twelve and seven are how many ? 26. Twelve and eight are how many ? 27. Twelve and nine are how many ? 28. Twelve and ten are how many ? 29. Thirteen and two are how many? 30. Thirteen and three are how many ? 31. Thirteen and four are how many ? 32. Thirteen and five are how many ? 33. Thirteen and six are how many ? 34. Thirteen and seven are how many ? 35. Fourteen and two are how many ? 36. Fourteen and three are how many ? 37. Fourteen and four are how many ? 38. Fourteen and five are how many ? 39. Fourteen and six are how many ? 40. Fifteen and two are how many ? 41. Fifteen and three are how many ? 42. Fifteen and four are how many ? 43. Fifteen and five are hew many? 44. Sixteen and two are how many ? 45. Sixteen and three are how many ? 46. Sixteen and four are how many ? 47. Seventeen and two are how many ? 28 ARITHMETIC. [Part 1. 48. Seventeen and three are how many ? 49. Eighteen and two are how many? E. 1. A man bo ight fx sheep for nine dollars, and to pay for it he gave live bushels of corn worth four dollars, and the rest in money ; how much money did he pay ? 2. If a barrel of flour is worth eight dollars, and a hundred-Weight of sugar is worth twelve dollars; how much more is the sugar worth than the Hour ? 3. If a man had eleven dollars, and should buy three bushels of corn for five dollars, how much money would he have left ? 4. A man bought a firkin of butter for fifteen dol- lars, but it being damaged, lie was willing to sell it again for eight dollars less than he gave for it; what did he sell it for ? 5. A man bought three barrels of flour for eighteen dollars, and sold it again for eleven dollars ; what did he lose by the bargain ? 6. A man bought a keg of tobacco for thirteen dol- lars, and sold it again for eighteen ; what did he gain by the bargain ? 7. Five less two are how many ? 8. Seven less three are how many ? 9. Three less three are how many ? 1 0. Nine less three are how many ? 11. Six less two are how many ? 12. Seven less four are how many ? 13. Eight less three are how many ? 14. Five less four are how many ? 1 5. Seven less five are how many ? 16. Nine less five are how many ? 17. Eight less six are how many ? 1 8. Eleven less two are how many ? 19. Twelve less four are how many? Sect. 1] ARITHMETIC. M 20. Ten less seven are how many ? 21. Thirteen less five are how many ? 22. Fourteen less eight are how many ? 23. Twelve less seven are how many ? 24. Seventeen less five are how many ? 25. Eighteen less ten are how many ? ^26. Thirteen less seven are how many ? 27. Sixteen less seven are how many ? 28. Fifteen less seven are how many ? 29. Nineteen less six are how many ? 30. Eighteen less five are how many ? 31. Seventeen less eight are how many ? 32. Fourteen less nine are how many ? 33. Sixteen less five are how many ? 34. Fifteen less eight are how many ? 35. Fourteen less nine are how many ? 36. Sixteen less ten are how many? 37. Seventeen less nine are how many ? 38. Eighteen less seven are how many ? F. 1. How many are nine and two ? Nineteen and two ? Twenty-nine and two ? Thirty-nine and two ? Forty-nine and two ? Fifty-nine and two ? Sixty-nine and two ? Seventy-nine and two ? Eighty-nine arid two ? Ninety-nine and two ? 2. How many are nine and three? Nineteen and three ? Twenty-nine and three ? Thirty-nine and hree ? Forty-nine and three ? Fifty-nine and three ? Sixty-nine and three ? Seventy-nine and three ? Eighty-nine and three? Ninety- .me and three ? 3. How many are nine and four ? Nineteen and four? Twenty-nine and four? .thirty-nine and four? Forty-nine and four ? Fifty-nine and four ? Sixty- aine and four ? Seventy-nine and four ? Eighty-nine and four ? Ninety-nine and four ? 4. How many are nine and five ? Nineteen and Gve ? Twenty-nine and five ? Thirty- nine and five ? 3U ARITHMETIC. [Pan t Forty-nine and five ? Fifty-nine and five ? Sixty-nine and live ? Seventy-nine and five ? Eighty-nine and five ? Ninety-nine and five ? 5. How many are nine and six? Nineteen and six ? Twenty-nine and six ? Thirty-nine and six : Forty-nine and six? Fifty-nine and six '{ Sixty-nine and six? Seventy-nine and six? Eighty-nine and six ? Ninety-nine and six ? 6. How many are nine and seven ? Nineteen and seven ? Twenty-nine and seven ? Thirty-nine and seven ? Forty-nine and seven ? Fifty-nine and seven ? Sixty-nine and seven ? Seventy-nine and seven ? Eighty-nine and seven? Ninety-nine and seven? 7. How many are nine and eight ? Nineteen and eight? Twenty-nine and eight? Thirty-nine and eight? Forty-nine and eight? Fifty-nine and eight? Sixty-nine and eight? .Seventy-nine and eight? Eighty-nine and eight ? Ninety-nine and eight ? 8. How many are nine and nine ? Nineteen and nine ? Twenty-nine and nine ? Thirty-nine and nine ? Forty-Bine and nine? Fifty-nine and nine ? Sixty-nine and nine ? Seventy-nine and nine? Eighty-nine and nine ? Ninety-nine and nine ? 9. How many are nine and ten ? Nineteen and ten? Twenty-nine and ten ? Thirty-nine and ten ? Forty- nine and ten ? Fifty-nine and ten ? Sixty-nine and ten? Seventy-nine arid ten? Eighty-nine and ten? Ninety-nine and ten ? 10. How many are eight and three? Eighteen and tliree ? Tweiu '-eight and three ? Tliirty-eight and three? Forty-e* lit and three? Fifty-eight and tliree ? Sixty-eight a,ad three ? Seventy-eight and three ? Eighty-eight and tliree ? Ninety-eight and three ? 11. How many are eight and four? Eighteen and four ? Twenty-eight and four ? Thirty-eight and lour ? v orty-eight and four? Fifty-eight and four? Sixty- Sect. 1.] ARITHMETIC. 31 eight and four ? Seventy-eight and four ? Eighty-eight and four ? Ninety -eight and four ? 12. How many are eight and five? Eighteen and five ? Twenty-eight and five ? Thirty-eight and five ? Forty-eight and five ? Fifty-eight and five ? Sixty- eight and five ? Seventy-eight and five ? Eighty-eight and five ? Ninety-eight and five ? 13. How many are eight and six? Eighteen and six ? Twenty-eight and six ? Thirty-eight and six ? Forty-eight and six ? Fifty-eight and six ? Sixty-eight and six ? Seventy-eight and six ? Eighty-eight and six? Ninety-eight and six? 14. How many are eight and seven ? Eighteen and seven? Twenty-eight and seven? Thirty-eight and seven ? Forty-eight and seven ? Fifty-eight and seven ? Sixty-eight and seven ? Seventy-eight and seven ? Eighty-eight and seven ? Ninety-eight and seven ? 15. How many are eight and eight? Eighteen and eight ? Twenty-eight and eight ? Thirty-eight and eight ? Forty-eight and eight ? Fifty-eight and eight ? Sixty-eight and eight ? Seventy-eight and eight ? Eighty-eight and eight ? Ninety -eight and eight ? 1 6. How many are eight and nine ? Eighteen and nine ? Twenty-eight and nine ? Thirty-eight and nine ? Forty-eight and nine ? Fifty-eight arid nine ? Sixty- eight and nine ? Seventy-eight and nine ? Eighty- eight and nine ? Ninety-eight and* nine ? 17. How many are seven and four i Seventeen and four ? Twenty-seven and four ? Thirty-seven and four ? Forty-seven and four ? Fifty-seven and four ? Sixty-seven and four ? Seventy-seven and four? Eighty-seven and four? Ninety-seven and four ? 18. How many are seven and five ? Seventeen and five ? Twenty-seven and five ? Thirty-seven and five Forty-seven and five ? Fifty-seven and five ? Sixty 32 ARITHMETIC. [Part 1 seven and five? Seventy-seven and five? Eighty- seven and five ? Ninety-seven and five ? 19. How many are seven and six ? Seventeen and six? Twenty-seven and six? Thirty-seven and six? Forty-seven and six ? Fifty-seven and six ? Sixty- seven and six? Seventy-seven and six? Eighty-seven and six ? Ninety-seven and six ? 20. How many are seven and seven ? Seventeen and seven ? Twenty-seven and seven ? Thirty-seven and seven ? Forty-seven and seven ? Fifty-seven and seven ? Sixty-seven and seven ? Seventy-seven and seven? Eighty-seven and seven ? Ninety-seven and seven ? 21. How many are seven and eight ? Seventeen and eight? Twenty-seven and eight? Thirty-seven and eight ? Forty-seven and eight ? Fifty-seven and eight ? Sixty-seven and eight ? Seventy-seven and eight? Eighty-seven and eight? Ninety-seven and eight? '2'2. How many are seven and nine? Seventeen and nine ? Twenty-seven and nine ? Thirty-seven and nine ? Forty-seven and nine ? Fifty -seven and nine ? Sixty-seven and nine ? Seventy-seven and nine ? Eighty-seven and nine ? Ninety-seven and nine ? 23. How many are six and five ? Sixteen and five ? Twenty-six and five ? Thirty-six and five ? Forty-six and five ? Fifty-six and five ? Sixty-six and five ? Seventy-six and fiVe ? Eighty-six and five ? Ninety- six and five ? 24. How many are six and six ? Sixteen and six ? Twenty-six and six ? Thirty-six and six ? Forty-six and six ? Fifty-six and six ? Sixty-six and six ? Sev- enty-six and six ? Eighty-six and six ? Ninety-six and six ? 25. How many are six and seven? Sixteen and seven? Twenty-six and seven ? Thirty-six and seven ? Forty-six and seven ? Fifty-six and seven ? Sixty -six Sect, i.j ARITHMETIC. 33 and seven ? Seventy-six and seven ? Eighty-six and seven ? Ninety-six and seven ? 26. How many are six and eight? Sixteen and eight ? Twenty -six and eight ? Thirty-six and eight ? Forty-six and eight ? Fifty-six and eight ? Sixty-six and eight ? Seventy-six and eight ? Eighty-six and eight ? Ninety-six and eight ? 27. How many are six and nine? Sixteen and nine ? Twenty-six and nine ? Thirty-six and nine ? Forty-six and mne ? Fifty-six and nine? Sixty-six and nine ? seventy-six and nine ? Eighty -six and nine ? Ninety-six and nine ? 28. How many are five and six? Fifteen and six? Twenty-five and six ? Thirty-five and six ? Forty- five and six ? Fifty-five and six ? Sixty-five and six ? Seventy-five and six ? Eighty-five and six ? Ninety- five and six ? 29. How many are five and seven ? Fifteen and seven ? Twenty-five and seven ? Thirty-five and seven ? Forty-five and seven ? Fifty-five and seven? Sixty-five and seven ? Seventy-five and seven ? Eigh- ty-five and seven ? Ninety-five and seven ? 30. I low many arc five and eight ? Fifteen and eight ? Twenty-five and eight ? Thirty-five and eight? Forty-five and eight? Fifty-five and eight? Sixty-five and eight ? Seventy -five and eight? Eigh- ty-five and .eight ? Ninety-five and eight ? 31. How many are five and nine ? Fifteen and nine? Twenty-five and nine? Thirty-five and nine? Pbrty-five and nine ? Fifty-five and nine ? Sixty-five and nine ? Seventy-five and nine ? Eighty-five and nine ? Ninety-five and nine ? 32. How many are four and seven ? Fourteen and seven ? Twenty-four and seven ? Thirty-four and seven ? Forty-four and seven ? Fifty-four and seven ? Sixty-four and seven ? Seventy-four and seven ? Eighty-four and seven ? Ninety-four and seven ? 3 34 ARITHMETIC. \Part 1 33. How many are four and eight ? Fourteen and eight ? Twenty-four and eight ? Thirty-four and eight ? Forty-four and eight ? Fifty-four and i-ight ? Sixty-four and eight ? Seventy-four and eight ? Eigh- ty-four and eight ? Ninety-four and eight ? 34. How many are four and nine? F'ourteen and nine ? Twenty-four and nine ? Thirty-four and nine ? Forty-four and nine ? Fifty-four and nine ? Sixty-four and nine ? Seventy-four and nine ? Eighty-four and nine ? Ninety-four and nine ? 35. How many are three and eight ? Thirteen and eight? Twenty-three and eight? Thirty- tliree and eight ? Forty-three and eight ? Fifty-three and eight ? Sixty-three and eight ? Seventy-three and fijrht ? Eighty-three and eight ? Ninety-three and eight ? 36. How manj are three and nin^ ? Thirteen and dine ? Twenty-three and nine ? Thirty-three and nine? Forty-three and nine? Fifty-three and nine ? Sixty-three and nine ? Seventy -three and nine ? Eigh- ty-three and nine ? Ninety-three and nine ? 37. How many are two and nine ? Twelve and nine ? Twenty-two and nine ? Thirty-two and nine ? Forty-two and nine ? Fifty-two and nine ? Sixty -two ami nine ? Seventy-two and nine ? Eighty-two and nine ? Ninety-two and nine ? G. 1. A man bought a firkin of butter for nine dollars, a keg of molasses for six dollars, and five bushels of wheat for seven dollars ; how much did he give for the whole ? 2. A boy gave some apples to his companions : to one he gave seven, to another six, and to another liijrht ; how many did he give to the whole ? 3. A man bought a cow for seventeen dollars, a sheep for nine, and a calf for seven ; how much did hi give for the whole ? Sect. 1.] ARITHMETIC. 35 4. A drover bought sheep as follows: of one man he bought twenty-seven, of another eight, of another ten, HIH! of another five ; afterwards he sold nine of them ; how many had he then ? 5. A lady bought a comb for thirty-seven cents > some tape for eight cents, some pins for ten cents, some needles for six cents, and some thread for six cents ; she gave seventy -five cents ; how much change ought she to receive back? 6. Eight, and nine, and six, are how many? 7. Five, and seven, and three, are how many ? 8. Four, and three, less two, are how many ? 9. Seven, and five, less three, are how many ? 10. Sixteen, and nine, and three, are how many ? 11. Twenty-three and eight are how many ? 12. Twenty-seven and five are how many ? 13. Twenty-five, less eight, are how many ? 14. Thirty-two and seven, less nine, are how many ? 15. Thirty-eight, and six, and four, less seven, are how many ? 1 6. Forty-four, and eight, and three, and seven, are how many? 17. Fifty-two, and six, and four, and five, and three, are how many ? 18. Fifty-seven, and six, and three, and five, and two, less eight, are how many? 19. Sixty-three, and five, and four, and six, and two, less seven, are how many ? 20. Seventy-five, and six, and eight, and three, and seven, and four, less nine, are how many ? 21. Eighty-three, and six, and five, and two, and *even, and nine, less four, are how many ? 22. Fifty-eight, and ten, and five, and seven, and three, and six, and four, less nine, are how many ? 23. Sixty-seven, and five, and eight, and nine, and seven, less six, are how many ? 24. Severity-four, and nine, arid seven, and five, and two, less six, are how many ? 36 ARITHMETIC. [Part L 25. Seventy-eight, and seven, and six, and two, and five, and eight, less nine, are how many ? 26. Eighty-four, and seven, and six, and eight, and five, less ton, are how many? 27. Forty-seven, and eight, and six, and two, and four, and eiirht, and three, and seven, and ten, and nine, less five, are how many? 28. Thirty-five, and eight, and four, and six, and three, and four, less eleven, are how many ? 29. Seventy, and ten, and six, and nine, and seven, and two, and five, and eight, and nine, less three, are how many ? H. 1. A man bought a cow for twenty-eight dol- lars, and a sheep for four dollars, and a pig for seven dollars ; how much did he give for the whole ? 2. James had twenty-seven cents ; John gave him four more, David seven, and George eleven, and he bought nine cents' worth of cake ; how many cents had he left? 3. A man paid sixteen dollars to A, nine dollars to B, seven dollars to C, ten dollars to D, six dollars to E, four dollars to F, and had eight dollars left ; how many had he at first ? 4. From Boston to Roxbury it is three miles ; from Roxbury to Dedham, six miles ; from Dedham to Walpole,. eleven miles ; from Walpole to Wrentham, four miles ; from Wrentham to Attlehorough, four miles ; from Attleborough to Pawtucket, nine miles ; from Pawtucket to Providence, four miles ; how many miles is it from Boston to Providence ? 5. One boy had fifteen nuts ; another boy gave him seven ; another, nine ; and another gave him enough to make his number forty ; how many did the last boy give him ? 6. A boy had thirty-seven apples ; he gave five to one companion ; and eight to another ; and when he Sect. 2.] ARITHMETIC. 37 had given some to another, he had six left ; how many did lie give to the last ? 7. A man owed fifty-six dollars ; at one time h paid seventeen dollars ; at another, eight ; at another, five ; at another, seven ; at last he paid the rest of the debt, wanting four dollars ; how much was the last payment ? 8. Six men bought a horse for seventy dollars ; the first gave twenty-three dollars ; the second, fifteen , the third, twelve ; the fourth, nine ; the fifth, se^en ; how much did the sixth give ? 9. A man bought a horse for forty-five dollars and paid fifteen dollars for keeping him ; he let him enough to receive twenty dollars ; and then sold him for forty- three dollars ; did he gain or lose by the bargain ? and how much ? SECTION H. A. 1. WHAT cost three yards of tape, at two cents a yard ? * 2. What cost four apples, at two cents apiece ? 3. What cost five peaches, at three cents apiece ? 4. What must you give for two oranges, at six cents apiece ? 5. What would be the price of three barrels of cider, at three dollars a barrel ? 6. If one orange is worth three apples, how many apples are four oranges worth ? 7. What are two barrels of flour worth, at five dollars a barrel ? * The pupil should be made to observe that three yards will cost three times as much as one yard ; and say, if one yard cost two cents, three yards will cost three times two cents. "He should be made to give this reason for the solution of each question, varying the numbers according to the question. 88 ARITHMETIC/ [Part 1 8. What cost three yards of cloth, at four dollars a yara ? 9. What cost two pounds of raisins, at eight centa a pound ? 10. What cost three lemons, at six cents apiece ? 11. If a man travel three miles in an hour, how many miles will he travel in four hours? 12. What will live pair of shoes come to at two dollars a pair ? 13. What is the price of seven yards of cloth, at three dollars a yard ? 14. What is the value of two pounds of beef, at seven cents a pound ? 15. If there are three feet in one yard, how many feet are there in four yards ? 1 6. How many feet are there in seven yards ? 17. How many feet are there in six yards and two feet? 18. If a man earn seven dollars in one week, how much would he earn in five weeks? 19. What cost seven hundred-weight of sugar, at nine dollars a hundred-weight ? 20. What cost seven pounds of sugar, at ten centa a pound ? 21. If one half yard of cloth cost three dollars, what would three yards cost ? 22. If one quarter of a yard of cloth cost two dol- lars, what is that a yard ? 23. How. many yards of cloth are there in seven pieces, each piece containing ten yards ? 24. What will live barrels of flour cost, at six dol- lars a barrel ? 25. If a man can travel four miles in an hour, how far can he travel in eight hours ? 26. If it take four bushels of wheat to make a bar- rel of flour, how many bushels will it take to make seven barrels ? Sect. 2.] ARITHMETIC. B. 1. Two times one are how many ? * 2. Two time's two are how many ? 3. Two times three are how many ? 4. Two times four are how many ? 5. Two times five are how many ? 6. Two times six are how many ? 7. Two times seven are how many 8. Two times eight are how many ? 9 Two times nine are how many ? 1 Two time? ten are how many ? 1 1. Three times one are how many ? 12. Three times two are how many ? 13. Three times three are how many ? 14. Three times four are how many ? 15. Three tunes five are how many ? 1 6. Three times six are how many ? 17. Three times seven are how many? 18. Three times eight are how many r Itf. Three times nine are how many ? 20. Three times ten are how many ? 21. Four times one are how many ? 22. Four times two are how many ? 23. Four times three are how many ? 24. F*our times four are how many ? 25. Four times five are how many ? 26. Four times six are how many ? 27. Four times seven are how many? 28. Four times eight are how many ? 29. Four times nine are how many ? 30. Four times ten are how many ? 31. Five times one are how many ? 32. Five times two are how many ? 83. Five times three are how many ? 34. F^ive times four are how many ? 35. Five times five are how many ? 36. Five times six are how many ? * See the Key. 40 ARITHMETIC. (Pan I 37. Five times seven are how many ? 38. Five times eight are how mariy ? 39. Five times nine are how many ? 40. Five times ten are how many ? 41. Six times one are how many ? 42. Six times two are how many ? 43. Six times three are how many ? 44. Six times four are how many ? 45. Six times five are how many ? 46. Six times six are how many ? 47. Six tunes seven are how many ? 48. Six times eight are how many ? 49. Six times nine are how many ? 50. Six times ten are how many ? 51. Seven times one are how many ? 52. Seven times two are how many ? 53. Seven times three are how many ? 54. Seven times four are how many ? 55. Seven times five are how many ? 56. Seven times six are how many ? 57. Seven times seven are how many ? 58. Seven times eight are how many ? 59. Seven times nine are how many ? 60. Seven tunes ten are how many ? 61. Eight times one are how many ? 62. Eight tunes two are how many ? 63. Eight times three are how many ? 64. Eight times *bur are how many ? 65. Eight times five are how many ? 66. Eight times six are how many ? 67. Eight times seven are how many ? 68. Eight times eight are how many ? 69. Eight times nine are how many ? 70. Eight times ten are how many ? 71. Nine times one are how many ? 72. Nine times two are how many ? 73. Nine times three are how many ? Sect. 8.] ARITHMETIC. 41 74. Nine times four are how many ? 75. Nine times five are how many ? 76. Nine times six are how many ? 77. Nine times seven are how many ? 78. Nine times eight are how many ? 79. Nine times nine are how many ? 80. Nine times ten are how many ? 81. Ten times one are how many ? 82. Ten times two are how many ? 83. Ten times three are how many ? 84. Ten times four are how many ? 85. Ten times five are how many ? 86. Ten times six are how many ? 87. Ten times seven are how many ? 88. Ten times eight are how many ? 89. Ten times nine are how many ? 90. Ten times ten are how many ? C. 1. Two times two are how many times one ? 2. Three times two are how many times one 3. Four times two are how many times one ? 4. Five times two are how many times one ? 5. Seven times two are how many ? 6. Nine times two are how many ? 7. Six times two are how many ? 8. Eight times two are how many ? 9. Ten times two are how many ? 10. Two times three are how many ? 11. Three times three are how many ? 12. Four times three are how many ? 13. . Five times three are how many ? 14. Six times three are how many ? 15. Eight times three are how many ? 16. Seven times three are how many ? 17. Ten times three are how many ? 18. Nine times three are how many ? 19. Two times four are how many ? 42 ARITHMETIC. 2u. Six times four are how many ? 21. Four times four are how many ? 22. Seven times four are how many ? 23. Nine times four are how many ? 24. Three times four are how many ? 2o. Five times four are how many ? 26. Ten times four are how many ? 27. Eight times four are how many? 28. Two times five are how many ? 29. Five times live are how many ? 30. Three times live are how many ? 31. Six times live are how many? 32. Two times six are how many? 33. Four times five are how many ? 34. Seven times five are how many ? 35. Three times six are how many ? 36. Seven times six are how many ? 37. Seven tunes seven are how many ? 38. Four times eight are how many ? 39. Six times seven are how many ? 40. Eight times nine are how many ? 41. Six times eight are how many? 42. Three times seven are how many ? 43. Four times nine are how many ? 44. Three times eight are how many? 45. Six times six are how many ? 46. Six times nine are how many ? 47. Nine times five are how many ? 48. Four times six are how many ? 49. Two times nine are how many ? 50. Seven times nine are how many ? 51. Nine times eight are how many ? 52. Two times eight are how many ? 53. Three times ten are how' many ? 54. Eight times seven are how many ? 55. Five times six are how many ? 56. Five times eight are how many ? ect. 2.] ARITHMETIC. 43 57. Two times seven are how many ? 58. Two times six are how many ? 59. Eight times six are how many ? 60. Four times seven are how many ? 61. Eight times eight are how many ? 6*2. Ten times five are how many ? 6.5. Seven times ten are how many ? 64. Ten times ten are how many ? 65. Nine times six are how many ? 66. Five times nine are how many ? 67. Three times nine are how many? 6se expressions ; the for- mer will be most easily understood by the \ upil ; it would be well, therefore, for the instructor to use the former frequently, though the latter is used for the most part in this treatise. 4 50 ARITHMETIC. [Part 1 34. Eleven are how many times three? 35. Twelve are how many times three? 36. If a yard of cloth be worth four dollars, and it be 'Hit into four equal parts, what will one of the parts be worth f that is, what is one fourth of it worth? What are two fourths of it worth ? What are three fourths of it worth ? 37. If you can buy a barrel of cider for four dollars, how much can you buy for one dollar ? How much for two dollars ? How much for three dollars? 38. What part of four is one ? Ans. One is the fourth part of four. 39. What part of four is two ? Ans. Two fourths of four. 40. What part of four is three ? Ans. Three fourths of four. 41. How many fourths make a whole one ? 42. If you can buy a bushel of corn for four shil- lings, how much can you buy for five shillings ? Ho\* much for six shillings ? JIow much for seven shil- lings? 43. Five are how many times four ? Ans. Once four and one fourth of four. 44. Six are how many times four ? Ans. Once four and two fourths of four. 45. Seven are how many times four ? Ans. Once four and three fourths of four. 46. Eight are how many times four ? 47. If four bushels of corn will buy one yard oi cloth, how many yards will nine bushels buy ? How many yards will ten bushels buy ? How many yards will eleven bushels buy ? 48. What do you understand by one fourth, tw fourths, or three fourths of anything ? See remark after example 16th. 49. Ten are how many times four ? 50. Eleven are how many times four ? Sect. 3.] ARITHMETIC. 51 51. Twelve are how many times four ? 52. Thirteen are how many times four ? 53. Fourteen are how many times four ? 54. Fifteen are how many times four ? 55. Sixteen are how many times four ? 56. If a barrel of flour be worth five dollars, and it be divided equally among five men, what will one man's share be worth ? that is, what is one fifth of a barrel worth ? What are two fifths of it worth ? What are three fifths of it worth ? What are four fifths of it worth ? 57. If five dollars will buy one box of butter, what part of a box will one dollar buy ? What part will two dollars buy ? What part will three dollars buy ? What part will four dollars buy ? 58. What part of five is one ? Ans. One is the fifth part of five. 59. Two is what part of five ? Ans. Two fifths of five. 60. Thr ;e is what part of five ? Ans. Three fifths of five. 61. Four is what part of five ? 62. How many fifths make a whole one ? 63. If cherries are five cents a quart, how many quarts can you buy for six cents ? How many for seven cents ? How many for eight cents ? How many for nine cents ? How many for eleven cents ? How many for thirteen cents ? 64. What do you understand by one fifth, two fifths &c. of anything ? See remark after example 16th. 65. Seven are how many times five ? Ans. Once five and two fifths of five ? 66. Eight are how many times five ? 67. Nine are how many times five ? 68. Ten are how many times five ? 69. Elevep are how many times five ? 52 ARITHMETIC. [Part 1 70. Twelve are how many times five ? 71. Thirteen are how many times five ? 72. Fourteen are how many times five ? 73. Fifteen are how many times five ? 74. If a barrel of beef cost six dollars, and it were divided into six equal parts, what would one of the parts be worth ? that is, what is one sixth of it worth ? What are two sixths of it worth ? What are three sixths of it worth ? Four sixths ? Five sixths ? 75. If fish is worth six dollars a ban-el, what part of a barrel will one dollar buy ? What part of a bar- rel will two dollars buy ? Three dollars ? Four dol- lars ? Five dollars ? 76. What part of six is one ? Am. One is the sixth part of six. 77. What part of six is two ? Ans. Two sixths of six. 78. Three is what part of six ? Ans. Three sixths of six. 79. Four is what part of six ? 80. How many sixths make a whole one ? 81. How much rye at six shillings a bushel can you buy for seven shillings ? How much for eight shillings ? Nine shillings ? Ten shillings ? Eleven shillings ? Twelve shillings ? Thirteen shillings ? Fifteen shil- lings ? Seventeen shillings ? 82. What do you understand by one sixth, two sixths, &c. 83. Eight are how many times six ? Ans. One time six and two sixths of six. 84. Nine are how many times six ? 85. Ten are how many times six ? 86. Eleven are how many times six ? 87. Twelve are how many times six ? 88. Thirteen are how many times six ? 89. Fourteen are how many times six ? 90. Fifteen are how many times six ? Sect. 3.] ARITHMETIC. *ft 91. If coal is worth seven dollars a chaldron, what is one seventh of a chaldron worth ? What are two sevenths of a chaldron worth ? Three sevenths ? Four sevenths ? Five sevenths ? Six sevenths ? 92. At the rate of seven dollars a yard, how much broadcloth can you buy for one dollar ? How much for two dollars ? How much for three dollars ? How much for four dollars ? How much for five dollars ? How much for six dollars ? How much for eight dol- lars ? How much for ten dollars ? How much for twelve dollars ? How much for fifteen dollars ? 93. What part of seven is one ? Ans. One is one seventh of seven. 94. What part of seven is two ? Ans. Two sevenths of seven. 95. What part of seven is three ? 9 6. Four is what part of seven ? 97. Five is what part of seven ? 98. What do you understand by one seventh, two sevenths, &c. of anything ? 99. How many sevenths make- a whole one ? 100. Nine are how many times seven ? 101. Ten are how many times seven ? 102. Eleven are how many times seven ? 103. Twelve are how many times seven ? 104. Thirteen are how many times seven? 105. Fourteen are how many times seven? 106. Fifteen are how many times seven ? 107. Sixteen are how many times seven? 108. When wheat is eight shillings a bushel, what is one eighth of a bushel worth? What are two eighths of a bushel worth ? What are three eighths of a bushel worth ? What are four eighths of a bushel worth ? Five eighths ? Six eighths ? Seven eighths ? 109. When wood* is eight dollars a cord, what part jf a cord can you buy for a dollar ? What part of a cord can you buy for two dollars ? What part for 54 ARITHMETIC. [Part I. three dollars ? What part for four dollars ? What part for five dollars ? What part for six dollars ? What part for seven dollars ? How much can you buy for nine dollars ? How much for ten dollars ? How much for eleven dollars ? How much for thirteen dollars ? How much for fifteen dollars ? How much for nine- teen dollars ? 110. What part of eight is one ? 111. What part of eight is two? 112. Three is what part of eight? 113. Four is what part of eight? 114. Five is what part of eight? 115. What do you understand by one eighth, two eighths, &c. of any number ? 116. Seven js what part of eight ? 117. How many eighths make a whole one ? 118. Ten are how many times eight ? 119. Eleven are how many times eight? 120. Twelve are how many times eight? 121. Thirteen are how many times eight? 122. Fourteen are how many times eight? 123. When sugar is nine dollars a hundred- weight, what is one ninth of a hundred-weight worth ? What are two ninths of a hundred-weight worth ? Three ninths ? Four ninths ? Five ninths ? Six ninths ? Seven ninths ? Eight ninths ? 124. When rye is nine shillings a bushel, what part of a bushel can you buy for one shilling ? What part for two shillings ? What part for three shillings ? For four shillings ? For five shillings ? For six shillings ? Seven shillings ? Eight shillings ? How much for ten shillings ? For thirteen shillings ? For fourteen shil- lings ? Sixteen shillings ? Twenty shillings ? 125. What do you understand by one ninth, twc ninths, three ninths, &c. of any number ? 126. Three is what part of nine ? 127. Four is what part of nine ? Jftwi. 3.] ARITHMETIC. 55 128. Five is what part of nine ? 129. Seven is what part of nine? 130. How many ninths make a whole one ? 131. Tim-teen are how many times nine? 132. Fifteen are how many times nine ? 133. Seventeen are how many times nine ? 134. When hay is ten dollars a ton, what is one tenth of a ton worth ? What are two tenths of a ton worth ? What are three tenths of a ton worth ? Four tenths ? Five tenths ? Six tenths ? Seven tenths ? Eight tenths ? Nine tenths ? 135. When sugar is ten dollars a hundred-weight, what part of a hundred-weight can you buy for one dollar ? What part for two dollars ? What part for three dollars ? What part for four dollars ? What part for five dollars ? Six dollars ? Seven dollars ? Eight dollars ? Nine dollars ? How much can you buy for eleven dollars ? Thirteen dollars ? Fifteen dollars ? Seventeen dollars ? 136. What do you understand by one tenth, two tenths, three tenths, &c. of anything ? * 137. How many tenths make a whole one ? * C. Instead of writing the names of numbers, it is usual to express them by particular characters, called figures. f One is written. --------I' Two is written 2 Three is written --------3 Four is written --- 4 Five is written -- - - - . - - - 5 Six is written ..----.-6 Seven is written ..... 7 Eight is written 8 Nine is written ------- - 9 Ten is written ---.-.--10 * These questions should frequently be put to the pupils. 56 ARITHMETIC. [Part 1 1. Eleven times one are how many times 2 ? 2. Twelve are how many times 2 ? 3 ? 4 ? 3. Fourteen are how many times 2 ? 4 ? 3 ? 4. If you had fifteen cents, how many cakes could you buy at 4 cents apiece ? How many at 2 cents apiece ? How many at 3 cents apiece ? How many at 5 cents apiece ? 5. Fifteen are how many times 4? 2? 3? 5? 6. Sixteen are how many tunes 5? 3? 6? 2? 7? 4? 7. Seventeen are how many times 6? 2? 7? 3? 5? 4? 8. Eighteen are how many times 4? 7? 9? 6? 3? 2? 5? 8? 9. Nineteen are how many times 3? 7? 4? 5? 8 ? 6 ? 9 ? 2 ? 10 ? 10. Twenty are how many times 6? 2? 8? 3? 9? 4? 10? 5? 7? 11. Twenty-one are how many times 7 ? 3 ? 8 ? 2? 4? 6? 9? 5? 10? 12. Twenty-two are how many times 3 ? 8 ? 5 ? 4?9?6?7?10?2? 13. If you had twenty-seven dollars, how much cloth could you buy at 9 dollars a yard ? How much at 6 dollars a yard ? How much at 4 dollars a yard ? How much at 3 dollars a yard ? How much at 7 dollars a yard ? How much at 8 dollars a yard ? How much at 5 dollars a yard ? How much at 10 dollars a yard ? 14. Twenty-seven are how many times 9 ? 6 ? 4 ? 3? 7? 8? 5? 10? 15. Twenty-four are how many times 6 ? 8 ? 7 ? 5 ? 2 ? 10 ? 3 ? 4 ? 9 ? 16. Twenty-nine are how many tunes 3 ? 7 ? 5 ? 9 ? 6 ? 8 ? 4 ? 10 ? 17. Twenty-three are how many times 4 ? 2 ? 7 ? 8? 3? 9? 6? 5? 10? Sect. 3.] ARITHMETIC. 57 18. Twenty-five are how many times 3 ? 7 ? 2 ? 6?9?4?8?5?10? 19. Thirty are how many times 10 ? 2 ? 3 ? 7 ? 9? 6? 5? 4? 8? 20. Thirty-three are how many times 6 ? 8 ? 7 ? 4 ? 9 ? 5 ? 10 ? 3 ? 21. Twenty-six are how many times 9? 4? 7? 3? 8? 5? 6? 10? 22. Thirty-five are how many times 5 ? 6 ? 3 ? 7? 9? 10? 4? 8? 23. Thirty-eight are how many times 8 ? 6 ? 3 ? 9? 5? 4? 7? 10? 24. Thirty-four are how many times 7 ? 3 ? 9 ? 10? 6? 8? 4? 5? 25. Thirty-six are how many times 8? 9? 4? 5? 3? 6? 7? 10? 26. Forty are how many times 8?10?6?4?3? 9? 5? 7? 27. For forty-seven cents, how many pounds of meat can be bought at 6 cents a pound ? How manj pounds at 8 cents ? How many at 9 cents ? How many at 3 cents ? How many at 5 cents.? How many at 4 cents ? How many at 7 cents ? How many at 10 cents a pound ? 28. Forty-seven are how many times 6 ? 8 ? 9 ? 3? 5? 4? 7? 10? 29. Forty-three are how many times 9 ? 8 ? 7 ? 6? 4? 3? 5? 10? 30. Forty-five are how many times 10 ? 8 ? 3 ? 6 ? 4? 7? 5? 9? 31. Forty-nine are how many times 6 ? 10 ? 5 ? 9? 4? 8? 7? 32. Fifty- three are how many times 8?5?6?4? 7? 9? 10? 33. Fifty-seven are how many times 9 ? 7 ? 10 ? 6 ? 5 ? 8 ? 4 ? *34. Fifty-five are how many times 6? 4? 8? 10? 9 ? 7 ? 5 ? 58 ARITHMETIC. [Par* I 35. Forty-eight are how many times 7 ? 5 ? 9 ? 4? 6? 8? 10? 36. Fifty-four are how many times 5?9?6?4? 7? 10? 8? 37. Forty-four are how many times 4 ? 6 ? 9 ? 7? 5? 8? 10? 38. Fifty-eight are how many times 7 ? 6 ? 8 ? 4 ? 9? 5? 10? 39. Forty-six are how many times 8 ? 10 ? 1 ? G ? 9? 7? 5? 40. Fifty are how many times 9 ? 5 ? 4 ? 10 ? 8 ? 6? 7? 41. Fifty-nine are how many times 4? 8? 7? 6? 10 ? 9 ? 5 ? 42. Sixty-four are how many times 7 ? 5 ? 8 ? 10 ? 6? 9? 43. Sixty-eight are how many times 6 ? 8 ? 9 ? 7 ? 10? 5? 44. Fifty-two are how many times 4 ? 6 ? 8 ? 10 ? 5? 7? 9? 45. Sixty-three are how many times 5? 4? 6? 10? 9? 7? 8? 46. Sixty-two are how many times 4 ? 10 ? 9 ? 7 ? 8? 5? 6? 47. Seventy-three are how many times 10 ? 7 ? 8? 6? 5? 9? 48. Seventy-five are how many times 7 ? 8 ? 10 ? 5? 6? 9? 49. If you had sixty-seven dollars, how many barrels of flour could you buy at 5 dollars a barrel ? How many at 7 dollars a barrel*? How many at 6 dollars a barrel ? How many at 8 dollars a barrel ? How many at 1 dollars a barrel ? How many at 9 dollars a barrel ? 50. Sixty-seven are how many times 5 ? 7 ? 6 ? 8? 10? 9? 51. Seventy-four are how many times 10 ? 7 ? 8 ? o ? 5 ? 9 ? Sect. S.] ARITHMETIC. 59 52. Sixty are how many times 9 ? 10 ? 6 ? 4 ? 7 ? 5? 8? 53. Seventy-two are how many times 5 ? 7 ? 6 f 8? 10? 9? 54. Sixty-five are how many times 5? 10? 8? 6? 7? 9? 55. Sixty-one are how many times 4? 5? 7? 6? 8? 10? 9? 56. Seventy-nine are how many times 10 ? 9 ? 8 ? 6? 5? 57. Seventy are how many times 9? 5? 6? 8? 7? 10? 58. Eighty-two are how many times 10 ? 7 ? 8 ? 6? 9? 59. Sixty-six are how many times 9? 5? 6? 7? 10? 8? 60. Eighty are how many times 10? 7? 6? 8? 9? 61. Sixty-nine are how many times 9? 5? 7? 10? 8? 6? 62. Eighty-one are how many times 10 ? 6 ? 8 ? 7? 9? 63. Seventy-six are how many tunes 9 ? 5 ? 10 ? 6? 7? 8? 64. Eighty-three are how many times 10 ? 6 ? 7 ? 9? 8? 65. Seventy-one are how many times 9 ? 5 ? 7 ? 6? 8? 10? 66. Eighty-four are how many times 10 ? 6 ? 8 ? ? 7 ? 67. Seventy-seven, are how many times 9 ? 7 ? 5 10? 8? 6? 68. Eighty-five are how many times 10 ? 8 ? 7 ? 6? 9? 69. Ninety are how many times 9 ? 10 ? 6 ? 7 ? 8? 70. Eighty -six are how many times 10 ? 9 ? 6 ? 7? 8? 60 ARITHMETIC. [Part \ 71. Ninety-four are how many times 9 ? 10 ? 8 ? 6? 7? 72. Eighty-seven are how many times 10 ? 9 ? 7 ? 6? 8? 73. Ninety-two are how many times 9 ? 10 ? 6 ? 7? 8? 74. Eighty-eight are how many times 10 ? 9 ? 8 6? 7? 75. Ninety-five are how many times 9 ? 10 ? 6 ? 8? 7? 76. Eighty-nine are how many times 10? 9? 6? 7? 8? 77. Ninety-eight are how many times 10 ? 9 ? 8 ? 6? 7? 78., One hundred are how many times 6 ? 10 ? 8 ? 9? 7? 79. Ninety- three are how many times 10 ? 6 ? 9 ? 8? 7? 80. Ninety-nine are how many times 7 ? 10 ? 8 ? 6? 9? 81. Ninety-six are how many times 9? 8? 7? 6? 10? 82. Ninety-seven are how many times 10 ? 9 ? 6 ? 7? 8? D. 1. If an orange is worth 3 apples, how many oranges are fifteen apples worth ? 2. In 8 pints how many quarts ? 3. In 8 gills how many pints ? 4. If you divide twelve apples equally among three oys, how many would you give them apiece ? 5. How- many hours would it take you to travel 10 miles, if you travel three miles in an hour ? 6. How many pence are there in .eight farthings ? 7. How many pence are there in twelve farthings ? 8. How many pence are there in seventeen far- things ? Sect. 3.] ARITHMETIC. 61 9. How many gallons are there in ten quarts ? 10. How much broadcloth, at 6 dollars a yard, can you buy for seventeen dollars ? 11. How many pounds of raisins, at 8 cents a pound, can you buy for twenty-five cents ? 12. In twenty-eight shillings, how many dollars ? 13. In twenty-eight farthings, how many pence ? 14. How many barrels of flour, at 7 dollars a bar- tel, can you buy for thirty -four dollars ? 15. How many reams of paper, at five dollars a ream, can you buy for thirty-seven dollars ? 1 6. In thirty-four gills, how many pints ? 17. In twenty-seven quarts, how many gallons ? 18. If an orange is worth six apples, how many oranges can you buy for forty apples ? 19. Thirty-six shillings are how many dollars ? 20. A man bought thirty apples at the rate of 3 for a cent, how many cents did they come to ? 21. A laborer engaged to work 8 months for ninety-six dollars ; how much did he receive for a month ? how much a week, allowing 4 weeks to the month ? how many shillings a day, allowing 6 work- ing-days to the week ? 22. If wine is worth twenty cents a pint, what is 1 gill worth ? 23. If you can buy a bushel * of apples for forty cents, what is the price of a- peck ? 24. If you buy a bushel of pears for forty-eight cents, what will be the price of half a peck, at the same rate ? 25. Four men bought a horse for forty-eight dol- lars ? what did each man pay ? 26. Five men bought a horse for seventy-five dollars, and sold him again for forty dollars ; what iid each man lose by the bargain ? * 1 bushel is 4 pecks- 62 ARITHMETIC. [Part 1 27. A man gave sixty-three cents for a horse to ride nine miles ; what was the price of one mile's ride? 28. A man hired a horse to ride, and agreed to give 8 cents a mile ; he paid fifty-six cents ; how many miles did he ride ? 29. A man had forty-two dollars, which he paid for wood at 7 dollars a cord ; how many cords did he buy ? 30. Two boys are forty-eight rods apart, -and both running the same way ; but the hindermost boy gains upon the other 3 rods in a minute ; in how many minutes will he overtake the foremost boy ? 31. There is a vessel containing sixty-three gal- lons of wine ; it has a pipe which discharges 7 gallons in an hour ; how many hours will it take to empty the vessel ? 32. There is a vessel containing eighty-seven gal- lons, and by a pipe ten gallons will run into it in an hour ; in how many hours will the vessel be filled ? 33. If one man can do a piece of work in thirty days, in how many days can 3 men do it ? in how many days can 5 men do it ? 34. If you wish to put sixty-four pounds of butter into 8 boxes, how many pounds would you put into each box ? 35. If you had seventy-two pounds of butter, which you wished to put into boxes containing 8 pounds each, how many boxes would it take. 36. If a man can perform a journey in thirty-sjx hours, how many days will it take him to do it when the days are nine hours long ? 37. If a man can do a piece of work in forty-eight hours, how many days would it take him to do it, if he works twelve hours in a day ? Sect. 4.J ARITHMETIC. 63 SECTION IV. A. 1. AT two cents a yard, what will 3 yards and one half of a yard of tape cost ? 2. 3 times 2, and one half of 2 are how many ? 3. At three dollars a yard, what will 4 yards and one third of a yard of cloth cost ? 4. 4 times 3, and one third of 3 are how many ? 5. At 3 dollars a barrel, what will 3 barrels and 2 thirds of a barrel of cider cost ? 6. 3 times 3, and two thirds of 3 are how many ? 7. If a man earn 4 dollars in a week, how many dollars will he earn in 3 weeks and 1 fourth of a week ? 8. 3 times 4, and 1 fourth of 4 are how many ? 9. If a yard of cloth cost 4 dollars, what will 5 yards and 3 fourths of a yard cost ? 10. 5 times 4, and 3 fourths of 4 are how many ? 11. If a man spend five dollars in a week, how many dollars will he spend in 3 weeks and 1 fifth of a week ? How much in 5 weeks and 2 fifths of a week ? 12. 3 times 5, and 1 fifth of 5 are .how many ? 13. 5 times o, and 2 fifths of 5 are how many ? 14. 6 times 5, and 3 fifths of 5 are how many ? 15. If beer is worth six dollars a barrel, what would 4 barrels and 1 sixth of a barrel cost ? How much would 7 barrels and 5 sixths of a barrel cost ? 1 6. 4 times 6, and 1 sixth of six are how many ? 17. 7 times 6, and 5 sixths of six are how many ? 18. At 7 dollars a barrel, what will 3 barrels and 1 seventh of a barrel of 'flour cost? What will 5 barrels and 2 sevenths of a barrel cost ? 19. 3 times 7, and 1 seventh of 7 are how many? 20. 5 times 7, and 2 sevenths of 7 are how many I 21. 8 times 5, and 4 fifths of 5 are how many ? 64 ARITHMETIC. [Par/ 1. 22. 8 times 6, and 3 sixths of 6 are how many ? 23. At 8 dollars a yard, what will 4 yards and 1 eighth of a yard of broadcloth cost ? 24. 4 times 8, and 1 eighth of 8 are how many ? 25. 2 times 7, and 3 sevenths of 7 are how many ? 26. 8 times 7, and 4 sevenths of 7 are how many ? 27. 9 times 7, and 6 sevenths of 7 are how many ? 28. 3 times 8, and 5 eighths of 8 are how many ? 29. 9 times 8, and 7 eighths of 8 are how many ? 30. If a hundred-weight of sugar cost 9 dollars, what will 2 hundred-weight and 1 ninth of a hundred- weight cost ? What will 5 hundred-weight and 2 ninths of a hundred-weight cost ? 31. 2 times 9, and 1 ninth of 9 are how many ? 32. 5 times 9, and 2 ninths of 9 are how many ? 33. 6 times 9, and 4 ninths of 9 are how many ? 34. 2 times 10, and 3 tenths of 10 are how many? 35. 7 times 9, and 7 ninths of 9 are how many ? 36. 5 times 10, and 4 tenths of 10 are how many ? 37. 8 times 9, and 5 ninths of 9 are how many ? 38. 4 times 10, and 7 tenths of 10 are how many? 39. 6 times 10, and 9 tenths of 10 are how many ? B. 1. A man bought 2 oranges at 6 cents apiece ; how many cents did they come to ? He paid for them with cherries at 4 cents a pint ; how many pints did it take ? 2. 2 times 6 are how many times 4 ? 3. A man bought 3 yards of cloth at 4 dollars a yard ; how many dollars did it come to ? How much flour at 6 dollars a barrel would it take to pay for it ? 4. 3 times 4 are how many times 6 ? 5* A man bought 4 peaches at 3 cents apiece ; how many cents did they come to ? He paid for them with pears at 2 cents apiece ; how many pears did it take? 6. 4 times 3 are how many times 2 ? ttcci. 4.] ARITHMETIC. 66 7. Bought 2 hundred- weight of sugar, at 9 dollar* a hundred-weight, and paid for it with wood at 6 dol- lars a cord ; how many cords did it take ? 8. 2 times 9 are how many times 6 ? 9. Bought 3 ban-els of flour at 8 dollars a barrel, and paid for it with cider at 4 dollars a barrel ; how many barrels did it take ? 10. 3 times 8 are how many times 4 ? 11. 12 times 3 are how many times 5 ? 12. 6 times 4 are how many times 8 ? 13. 3 times 10 are how many times 6 ? 14. 4 times 9 are how many times 6? 15. How much flannel worth 4 shillings a yard must be given for 3 yards of silk worth 5 shillings a yard ? 1 G. 3 times 5 are how many times 4 ? 17. 2 times 7 are how many times 3 ? 5 ? 4 ? 18. 4 times 5 are how many times 3 ? 6 ? 7 ? 19. 3 times 7 are how many times 4? 5? 6? 8? 9? 20. Bought 2 kegs and 2 sevenths of a keg ol tobacco at 7 dollars a keg, and paid for it with wood at 4 dollars a cord ; how many cords did it take ? How much butter at 3 dollars a box would it take to pay for it ? 21. 2 times 7, and 2 sevenths of 7 are how many times 4 ? 3 ? 5 ? 6 ? 8 ? 22. Bought 3 bushels and 3 fifths of a bushel of corn at 5 shillings a bushel, and paid for it with wheat at 6 shillings a bushel ; how many bushels of wheat did it take ? 23. 3 times 5, and 3 fifths of five are how many times 6 ? 9 ? 4 ? 7 ? 3 > 8 ? 24. How much sugar that is 8 dollars a hundred- * weight, can be bought for 4 cords and 2 sevenths of a cord of wood, at 7 dollars a cord ? 66 ARITHMETIC. [Part 1. 25. 4 times 7, and 2 sevenths of 7 are how many times 6?8?5?3?9?10? 26. 5 times 5, and 3 fifths of 5 are how many tunes 4?8?9?7?10?3?6? 27. 6 times 7, and 3 sevenths of 7 are how many times 9? 4? 5? 8? 10? 28. 5 times 8, and 3 eighths of 8 are how many times 6 ? 9 ? 4 ? 7 ? 10 ? 29. 7 times 8, and 5 eighths of 8 are how many times 9 ? 6 ? 10 ? 4 ? 5 ? 30. 5 times 9, and 4 ninths of 9 are how many times 7 ? 8 ? 6 ? 10 ? 4 ? 31. 7 times 9, and 7 ninths of 9 are how many times 6? 8? 10? 5? 4? 32. 6 times 10, and 3 tenths of 10 are how many times 7 ? 5 ? 4 ? 9 ? 8 ? 33. 8 tunes 10, and 4 tenths of 10 are how many times 6 ? 7 ? 9 ? 34. 8 times 9, and 3 ninths of 9 are how many times 6 ? 10 ? 7 ? C. 1. Bought 4 bushels of apples, at 3 shillings a bushel * how many dollars did they come to ? 2. How many apples, at 2 cents apiece, must you give for 2 lemons at 4 cents apiece ? 3. How many pears, at 3 cents apiece, must you give for 3 oranges at 5 cents apiece ? 4. How many barrels of cider, at 3 dollars a barrel, must be given for 5 boxes of butter, at 4 dollars a box? 5. A man bought 4 yards of broadcloth, at 7 dol- lars a yard, and paid for it with flour, at 5 dollars a barrel ; how many barrels did he give ? 6. If 2 apples cost 4 cents, what would 4 apples cost? 7. If 3 apples are worth 6 cents, how many apples Sect. 4.] AK1THMETIC. 67 must you give for 8 pears, that are worth 3 cents apiece ? 8. James had 8 oranges that were worth 5 cents apiece, and George had 5 quarts of cherries that were worth 6 cents a quart, which he gave to James for a part of his oranges : how many oranges did he buy, and how many had James left ? 9. Bought 8 yards of cloth, at 9 shillings a yard how many dollars did it come to ? 10. Bought 5 bushels and 3 sevenths of a bushel of salt, at 7 shillings a bushel ; how many dollars did it come to ? 11. Bought 9 boxes and 2 thirds of a box of raisins for 3 dollars a box, and paid for it with cider at 4 dol- lars a barrel ; how many barrels did it take ? 12. Bought 8 pounds and 4 sevenths of a pound of opium at 7 dollars a pound, and paid for it with cloth at 5 dollars a yard ; how many yards did it take? 13. Bought 6 chaldrons and 4 ninths of a chaldron of coal at 9 dollars a chaldron, and paid for it with oranges at 5 dollars a box ; how many boxes did it take? 14. Bought 7 cases and 5 sixths of a case of Flor- ence oil at 6 dollars a case, and paid for it with sheet- lead at 7 dollars a hundred-weight ; how many hun- dred-weight did it take ? 15. How many dozen of eggs, at 8 cents a dozen, must you give for 7 pounds of sugar, at 10 cents a pound ? 16. How much barley, at 3 shillings a bushel, must be given for 8 bushels of wheat, at 7 shillings a bushel ? 17. How much cloth, at 4 shillings a yard, must be given for a firkin of butter worth 8 dollars ? 18. How much cloth, at 5 shillings a yard, can be bought for 2 reams of paper, at 5 dollars a ream ? 68 ARITHMETIC. [Part 1 19. How much wheat, at 7 shillings a bushel, can be bought for 2 barrels of cider, at 4 dollars and a half a barrel ? 20. How long would it take a man to lay up 10 dollars, if he saves 4 shillings a day ? 21. If a man earns 8 shillings a day, how many dollars would he earn in 10 days ? 22. A man bought twenty pears at the rate of 2 for 8 cents ; how much did they come to ? 23. How many eggs, at the rate of 3 for 5 cents, can you buy for thirty cents? 24. A man hired a laborer and agreed to give him 5 dollars for every 3 days' work ; how much did he give him ' a week, there being 6 working days in a week ? How much was it a month, allowing 4 weeks to the month ? 25. If a man receives 5 dollars for 3 days* work, how many shillings is that a day ? 26. 5 men bought a horse lor sixty-three dollars, and paid two dollars a week for keeping him ; at the end of 8 weeks they sold him for fifty-four dollars ; how much did each man lose by the bargain ? SECTION V. A. 1. JAMKS had 4 apples, and John had half as many ; how many had he ? 2. If an orange cost 6 cents, and an apple half as much ; how much does the apple cost ? 3. If you divide 8 apples equally between two boys, what part of them must each have ? "Arts. One half of them. 4. What is 1 half of 8 ? 5. If you divide 8 apples equally among 4 boys what part of them must each have ? Ans. One fourth of them. Sect. 5.] ARITHMETIC. 69 6. What is 1 fourth of 8 ? 7. If you divide 6 oranges equally among 3 boys, what part of them must 1 boy have ? 8. What is 1 third of 6. 9. If 4 yards of cloth cost 8 dollars, what part of 8 dollars would 1 yard cost ? What part of 8 dollars would 2 yards cost ? What ,part of 8 dollars would 3 yards cost ? 10. What is 1 fourth of 8 ? What is 2 fourths of 8 ? What is 3 fourths of 8 ? 11. If 6 yards of cloth will make 3 coats, what part of 6 yards will make 1 coat ? What part of 6 yards will make 2 coats ? 12. What is one third of 6 ? What is two thirds of 6? 13. If 3 barrels of cider cost 9 dollars, 1 what part of 9 dollars will 1 barrel cost ? What part of 9 dol- lars will 2 barrels cost ? 14. What is 1 third of 9 ? What is 2 thirds of 9 ? 15. If 2 yards of cloth cost 10 dollars, what part of 10 dollars will 1 yard cost? What part of 10 dol- lars will 3 yards cost ? 16. What is 1 half of 10 ? What is 3 halves of 10 ? 17. If 2 barrels of flour cost twelve dollars, what part of twelve dollars will 1 barrel cost ? What part of twelve dollars will 3 barrels cost ? What part of twelve dollars will 5 barrels cost ? 18. What is 1 half of twelve ? What is 3 halves of 12 ? What is 5 halves of twelve ? 19. If 4 barrels of cider cost twelve dollars, what part of twelve dollars will 1 ban-el cost ? What part of twelve dollars will 3 barrels cost ? What part of twelve dollars will 5 barrels cost ? What part of twelve dollars will 7 barrels cost ? 20. What is 1 fourth of twelve ? What is 2 fourths of twelve ! 3 fourths ? 5 fourths ? 7 fourths ? 21. If 3 oranges cost twelve cents, what part of 70 AKITHMETICv [Port 1 twelve cents will 1 orange cost ? What part of twelve cents will 2 oranges cost ? What part of twelve cents will 4 oranges cost? What part of twelve cents will 5 oranges cost ? 7 oranges? 10 oranges ? 22. What is 1 third of twelve ? 2 thirds? 4 thirds? 5 thirds? 7 thirds? 10 thirds? 23. If 5 bushels of wheat cost 10 dollars, what part of 10 dollars will 1 bushel cost ? What part of 10 dollars will 2 bushels cost? 3 bushels? 4 bushels? 6 bushels ? 7 bushels ? 24. What is 1 fifth of 10? 2 fifths? 3 fifths? 4 fifths ? 6 iifths ? 7 fifths ? 25. What is 1 half of fourteen ? 1 seventh ? 2 sevenths ? 3 sevenths ? 5 sevenths ? 26. What is 1 third of fifteen? 2 thirds? 1 fifth? 2 fifths ? 3 fifths ? 4 fifths ? 27. What is 1 half of sixteen? 3 fourths? 1 eighth? 8 eighths ? 5 eighths ? 7 eighths ? 1 sixteenth ? 28. What is 1 half of eighteen ? 2 thirds? 1 sixth? 5 sixths? 1 ninth? 2 ninths? 4 ninths? 5 ninths:' 7 ninths ? 8 ninths ? 1 eighteenth ? 5 eighteenths ? 29. What is 1 half of twenty ? 1 fourth? 3 fourths? 1 fifth? 3 fifths?' 2 fifths? 4 fifths? 1 twentieth? 3 twentieths ? 7 twentieths ? 30. What is 1 third of twenty-one ? 2 thirds ? i seventh r 5 sevenths ? 3 sevenths ? 6 sevenths ? 4 sevenths ? 31. What is 1 half of twenty-two? 1 eleventh? 3 elevenths? 5 elevenths? 7 elevenths? 10 elev- enths ? 3*2. What is 1 half of twenty-four ? 1 third ? 2 thirds ? 1 fourth ? 3 fourths ? 1 sixth ? 5 sixths ? 5 eighths? 3 eighths? 7 eighths? 1 twelfth? 5 twelfths? 7 twelfths? 33. What is 1 half of twenty-six? 34. What are 2 thirds of twenty-seven? 2 ninths? 4 ninths ? 5 ninths ? 8 ninths ? 7 ninths ? Sect. 5.] ARITHMETIC. 71 35. What is 1 half of twenty-eight ? 3 fourths ? 2 seventlis ? 5 sevenths ? 3 sevenths ? 36. What is 1 sixth of thirty? 3 fifths? 7 tenths ? 2 thirds? 1 half? 37. What is 1 half of thirty-two ? 3 fourths ? 5 eighths ? 1 sixteenth ? 38. What is 1 half of thirty-four ? 39. What is 5 sixths of thirty-six ? 2 thirds ? 1 half? 7 twelfths? 4 ninths? 3 fourths? 40. What is 1 half of thirty-eight ? 41. What is 1 third of thirty-nine ? 42. What is 3 fourths of forty ? 7 eighths ? 9 tenths? 1 half? 43. What is 3 sevenths of forty-two ? 5 sixths ? B. 1. A boy having twelve apples, kept 1 fourth of them himself, and divided the other 3 fourths of them equally among 4 of his companions, how many did he give them apiece ? 2. 3 fourths of twelve are how many times 4 ? 3. A man having fourteen bushels of grain, divided 5 sevenths of it equally among 3 men ; how much did he give them apiece ? 4. 5 sevenths of fourteen are how many times 3 ? 5. A man having fifteen shillings, gave away 4 fifths of it, how many dollars did he give away ? 6. 4 fifths of fifteen are how many times 6 ? 7. A man having twenty-one cents, paid away 6 sevenths of them for oranges at 5 cents apiece ; how many oranges did he buy? 8. 6 sevenths of twenty-one are how many times 5? 9. A man bought a piece of cloth for twenty-four shillings, and sold it again for 5 thirds of what he gave for it ; how many dollars did he sell it for ? 10. 5 thirds of twenty-four are how many times 6? 11. 7 filths of twenty are how many times 6 ? 1*1 ARITHMETIC [Part 1 12 5 sevenths of thirty-five are how many times 8? 13. 7 sixths of thirty-six are how many times 5 ? 14. 9 sevenths of forty-two are how many times 8! 15. 7 ninths of forty-five sire how many times 6? 10. 5 sixths of forty-eight arc 4 how many times 7? 17. 6 fifths of fifty are how many times V ? 18. 8 ninths of fifty-four arc how many times 5? 19. 9 r^venths of fifty-six are how many times 10? 20. 7 sixths of sixty are how many times 8 ? 2}. 9 sevenths of sixty-three are how many times 7? 22. 10 eighths of sixty-four are how many times 9 ? 23. 6 eighths of seVenty-two are how many times 7? 24. 4 sevenths of eighty-four are how many times 9? 25. 7 tenths of ninety are how many times 8? C. ). Charles had 6 apples, and gave 1 third of them to John : how many did he give him ? 2. Albert had 9 cents, and spent 2 thirds of them ; how many had he left ? 3. James had 10 pears, and gave 1 half of them to one of his companions, and 2 fifths of them to another ; how many did he give away? 4. If 3 yards of cloth cost 6 dollars, what is that a yard ? 5. If 4 yards of cloth cost twelve dollars, what will 2 yards cost ? 6. If 9 apples cost eighteen cents, what will apples cost ? 7. If 3 oranges cost 18 cents, what will 2 cost ? 8. James had twenty-five cents, and he gave 4 fifths Df them for 1 apples ; how much did he give for all the apples ? how much apiece ? 9. A man had thirty dollars, and gave 5 sixths of SecL 5.] ARITHMETIC. 73 them for 8 yards of cloth ; how much did he give a yard ? 10. A man had forty yards of cloth, and sold 3 fifths of it for twenty-four dollars ; what was that a yard ? 11. A man had forty-two barrels of flour, and sold 2 sevenths of it for 6 dollars a barrel ; how much did it come to ? 12. A boy had sixty-three nuts,. and divided 4 sev- enths of them equally among six of his companions how many did he give them apiece ? 13. If 4 yards of cloth cost twelve dollars, wha will 3 yards cost ? 14. If 5 oranges cost twenty-five cents, what will 3 cost? 15. If 3 oranges cost fifteen cents, what will 7 cost? 16. If 3 barrels of cider cost twelve dollars, what will 10 barrels cost ? 17. If 7 pounds of flour cost thirty-five cents, what will 9 pounds cost ? 18. If 5 firkins of butter cost forty dollars, what will 3 firkins cost ? 19. If 2 men can do a piece of work in six days, how long would it take 4 men to do the same work ? 20. If 6 men can do a piece of work in twelve days* in how many days will 3 men do the same work ? 21. If 3 men can do a piece of work in twelve days, in how many days will 4 men do the same work ? 22. If 2 pipes of a certain size will empty* a cistern in 6 hours, in how long a time will 3 pipes of the same size empty it ? "23. Three men, setting out on a journey, purchased 5 loaves of bread apiece, but before they had eaten any of it, two other men joined them, and they agreed to share the bread equally among the whole ; how many loaves did they have apiece ? 74 ARITHMETIC. * [Part 1 24. If 4 barrels of flour cost twenty-four dollars what would 7 barrels cost ? How much cider at 3 dollars a barrel would 7 barrels of flour buy? 25. A man bought a quantity of flour for fifty-four dollars ; and another man gave him 9 yards of cloth for 5 sixths of it ; what was the cloth worth a yard ? 2G. If 9 yards of cloth cost fifty-four dollars ; how many boxes of butter, at 4 dollars a box, would 5 yards of the same cloth buy ? 27. Bought 8 firkins of butter for seventy-two dol- lars, and gave 6 of them for 7 yards of cloth ; what was a yard of the cloth worth ? 28. A man bought 6 barrels of flour for sixty dol- lars, and gave 4 barrels of it for cider at 5 dollars a Darrel ; how many barrels did he buy ? Note. The manner of writing numbers with figures has been explained as far as ten. The numbers from ten to one hundred are written as follows : Ten is written 10 Eleven x 11 Twelve 12 Thirteen 13 Fourteen ----------14 Fifteen 15 Sixteen 16 Seventeen 17 Eighteen ----------18 Nineteen ----------19 Twenty 20 Twenty-one 21 Twenty-two ---- 22 Twenty-three 23 Twenty-four ---------24 Twenty-five 25 Twenty-six - - 26 Twenty-seven --------27 Twenty-eight - - - 28 Sect. 6.] ARITHMETIC. 75 Twenty-nine 29 Thirty 30 Thirty-one, &c. - - 31 Forty- - , 40 Fifty 50 Sixty 60 Seventy 70 Eighty 80 Ninety 90 One hundred 100 SECTION VI. A. 1. BOUGHT 1 half of a yard of cloth for 1 shil- ling, what would be the price of a yard at the same rate? 2. If 1 half pint of cherries cost 2 cents, what will a pint cost ? 3. If 1 fourth of a barrel of flour cost two dollars, what would a barrel cost ? 4. 2 is 1 half of what number ? 1 fourth of what number ? 5. If 1 third of a yard of cloth cost 2 dollars, what is that a yard ? 6. 2 is 1 third of what number ? 7. If 1 third of a yard of cloth cost 3 dollars, what is that a yard ? 8. 3 is 1 third of what number ? 9. If 1 fourth 'of a firkin of butter cost 3 dollars, what is that a firkin ? 10. 3 is 1 fourth of what number ? 11. A man bought 1 third of a bushel of wheat foi 4 shillings, what would a bushel cost at that rate ? 12. 4 is 1 third of what number ? 13. If a man can ride 2 miles in 1 fifth of an hour, how far can he ride in an hour ? 76 ' ARITHMETIC. [Par/ 1 14. 2 is 1 fifth of what number ? 15. A man, being asked the age of his eldest son, answered that bis youngest son, who was 3 years old, was just 1 fifth of the age of his eldest son ; how old was the eldest son ''. 16. 3 is 1 fifth of what number? 17. A man bought 1 sixth part of a hundred-weigh! of sugar for 2 dollars, what would a hundred-weight cost at the same rate ? 18. 2 is 1 sixth of what number? 19. Bought 1 fifth of a pound of starch for 5 cents, what was that a pound ? 20. 5 is 1 fifth of what number ? 21. Bought 1 fourth of a pound of aquafortis for 6 cents, what was that a pound ? 22. 6 is 1 fourth of what number ? 23. Bought 1 seventh of a hundred-weight of cocoa for 4 dollars, what would 1 hundred- weight cost at the same rate ? 24. 4 is 1 seventh of what number ? 25. 7 is 1 fifth of what number ? 26. 5 is 1 third of what number ? 27. 4 is 1 eiirhth of what number ? 28. 6 is 1 sixth of what number ? 29. 8 is 1 third of what number ? 30. 9 is 1 fourth of what number ? 31. 7 is 1 sixth of what number ? 32. 8 is 1 seventh of what number ? 33. 9 is 1 eighth of what number ? 34. 8 is 1 tenth of what number? 35. 7 is 1 ninth of what number ? 36. 6 is 1 fifth of what number ? 37. 10 is 1 seventh of what number ? B. 1. A man bought some linen and some cotton cloth : for the linen he gave 4 shillings a yard, which was twice as much as he gave for the cotton ; what did he give a yard for the cotton ? feet. 6.] ARITHMETIC. 77 / 2. 4 is 2 times what number ? 3. If 2 thirds of a yard of cloth cost 6 dollars what would 1 third cost ? _ 4. 6 is 2 times what number ? 5. If 3 fourths of a barrel of flour cost 6 dollars, what will 1 fourth of a barrel cost ? 6. 6 is three times what number ? 7. If 2 fifths of a pound of chocolate cost 8 cents, what would 1 fifth of a pound cost ? 8. 8 is 2 times what number ? 9. If 3 fifths of a pound of candles cost 9 cents, what will 1 fifth of a pound cost ? 10. 9 is 3 times what number ? 11. If 2 sevenths of a pound uf spermaceti candles cost 10 cents, what will 1 seventh of a pound cost ? 12. 10 is 2 times what number? 13. If 5 eighths of a pound of cotton cost 10 cents, what will 1 eighth cost ? 14. 10 is 5 times what number ? 15. If 2 thirds of a yard of cloth cost 4 'dollars, what will 1 third cost ? If 1 third of a yard cost 2 dollars, what will a yard cost ? * 16. If 4 is 2 thirds of some number, what is 1 third of the same number ? 2 is 1 third of what num- ber ? * Then 4 is 2 thirds of what ? 17. If 2 thirds of a barrel of flour cost 6 dollars, what will 1 third of a barrel cost ? If 1 third of a barrel cost 3 dollars, what will a barrel cost ? * 18. If* 6 is two thirds of some number, what is 1 third of the same number ? 3 is 1 third of what number ? Then^ 6 is 2 thirds of what ? 19. If 3 fourths of a bushel of wheat cost 6 shil- lings, what will 1 fourth of a bushel cost ? If 1 fourth of a bushel cost 2 shillings, what will a busheJ cost ? 20. If 6 is 3 fourths of some number, what is 1 * See this section, article A. 78 ARITHMETIC. [Part 1. fourth of the same number ? 2 is 1 fourth of what number ? Then 6 is 3 fburths of what ? 21. If 2 fifths of a gallon of wine cost 4 shillings, what will 1 fifth of a gallon cost ? If 1 fifth of a gal- lon cost 2 shillings, what will a gallon cost ? 22. If 4 is 2 fifths of some number, what is 1 fifth of the same number ? 2 is 1 fifth of what number ? Then 4 is 2 fifths of what ? 23. If 3 sevenths of a pound of tobacco cost 6 cents, .what will 1 seventh of a pound cost? If 1 seventh of a pound cost 2 cents, what will a pound cost? 24. If 6 is 3 sevenths of some number, what is 1 seventh of the same number ? 2 is 1 seventh of what number ? Then 6 is 3 sevenths of what ? 25. If 2 sevenths of a barrel of fish cost 4 dollars, what will 1 seventh of a barrel cost ? What will a barrel cost ? 26. 4 is 2 sevenths of what number ? 27. If 3 eighths of a pound of chocolate cost 6 cents, what will 1 eighth of a pound cost? What will a pound cost ? 28. 6 is 3 eighths of what number ? 29. If eight cents will buy 2 fifths of a pound of aquafortis, how many cents will buy a pound ? 30. 8 is 2 fifths of what number ? 31. A man bought -3 fourths of a hundred- weight of yellow ochre for 9 dollars ; what was that a hundred- weight ? 32. 9 is 3 fourths of what number ? 33. 8 is 4 ninths of what number ? 34. 9 is 3 tenths of what number ? 35. 10 is 5 sevenths of what number ? 36. 12 is 3 fifths of what number ? 37. 12^ is 4 ninths of what number ?. 38. 10'is 2 sevenths of what number? 39. 14 is 7 fifths of what number ? Sect. 6.] ARITHMETIC. 79 40. 15 is 3 elevenths of what number ? 41. 16 is 2 fifths of what number? 42. 18 is 6 tenths of what number? 43. 20 is 5 ninths of what number ? 44. 21 is 3 ninths of what number ? 45. 25 is 8 ninths of what number ? C. 1. If 5 eighths of a cask of claret wine cost 15 dollars, what is that a cask ? How much cider at 6 dollars a Barrel would it take to pay for a cask of the wine ? 2. 15 is 5 eighths of how many times 4 ? , 3. If 2 thirds of a pound of coffee cost 18 cents, how much would a pound cost ? How many oranges at 5 cents apiece, might be bought for a pound ? 4. 18 is 2 thirds of how many times 5 ? 5. A man bought 4 sevenths of a hundred-weight of sugar for 20 shillings, how many dollars would a hun- dred-weight come to at the same rate ? 6. 20 is 4 sevenths of how many times 6 ? 7. A man sold a cow for 21 dollars, which was only 7 tenths of what she cost him ; how much did she cost him ? When he bought her, he paid for her with cloth at 8 dollars a yard ; how many yards of cloth did he give ? 8. 21 is 7 tenths of how many times 8 ? 9. A man, being asked the age of his youngest son, answered, that the age of his eldest son was 24 years, which was 3 fifths of his own age ; and that his own age was 10 times as much as that 'of his youngest Bon ; what was his age ? and what was the age of hia youngest son ? , 10. 24 is 3 fifths of how many times 10 ? 11. 27 is 3 fifths of how many times 7 ? 12. 28 is 7 tenths of how many times 9 ? 13. 30 is 5 eighths of how many times 7 ? 14. 32 is 4 sevenths of how many times 6? 80 ARITHMETIC. [Part 1 15. 36 is 9 eighths of how many times 5 ? 1 6. 40 is 8 ninths of how many times 8 ? 17. 42 is 6 fifths of how many times 4 ? 18. 45 is 9 eighths of how many times 6 ? 19. 48 is 8 ninths of how many times 7 ? 20. 50 is 5 sevenths of how many times 8 ? 21. 54 is 9 sixths of how many times 7 ? 22. 56 is 7 ninths of how many times 10 ? 23. 60 is 10 sevenths of how many times 4? 24. 63 is 9 eighths of how many times 5 ? 25. 64 is 8 ninths of how many times 7 ? 26. 70 is 10 sevenths of how many times 8 ? 27. 72 is 9 fifths of how many times 6 ? 28. 80 is 10 thirds of how many times 4 ? 29. 80 is 8 fifths of how many times 6 ? D. 1. A boy gave away 4 cents, which was 1 third of all he had ; how many had he at first ? 2. A boy gave 5 apples to one of his companions, which was 1 fourth of what he had ; how many had he? 3. A man paid away 4 dollars, which was 2 thirds ot all the money he had ; how much had he ? 4. A man sold a watch for 18 dollars, which was 3 fourths ef what it cost him ; how much did it cost ? 5. A man sold a cow for 15 dollars, which was 3 fifths of what the cow cost ; how much did he lose by his bargain ? 6. A man bought 12 yards of cloth, and sold it for 54 dollars, which was 9 eighths of what it cost him ; what did it cost him a yard ? and how much did he gain by his bargain ? 7. There is a pole standing in the water, so that 10 feet of it is above the water, which is 2 thirds of the whole length of the pole ; how long is the pole ? 8. There is a pole 2 thirds under water, and 4 feet mt ; how long is the pole ? Sect. 6.] ARITHMETIC. 81 9. There is a pole 2 fifths under water, and 6 feet out of the water ; how long is the pole ? 10. There is an orchard, in which 3 sevenths of the trees bear cherries, and 2 sevenths bear peaches, and 10 trees bear plums ; how many trees are there in the orchard ? and how many of each sort ? 11. There is a school, in which 2 ninths of the boys learn arithmetic, 3 ninths learn grammar, I ninth learn geography, 1 ninth learn geometry, and 1 2 learn to write ; how many are there in the school ? and how many attending to each study ? 12. A man sold a watch for 63 dollars, which was 7 fifths of what it cost him ; how much did he gain by the bargain ? Miscellaneous Examples. 1. IF 1 yard of cloth cost 4 dollars, what will 5 yards cost ? 2. A man bought 3 pounds of raisins, at 7 cents a pound, and 16 oranges, at 4 cents apiece, and 1 pound of candles for 1 6 cents ; what did they all come to ? 3. A boy had 37 apples ; he kept 5 himself, and divided the rest equally among four companions ; how many did he give them apiece ? 4. Two men are 40 miles apart, and both travelling the same way ; the hindermost man gains upon the other 5 miles each day ; in how many days will he overtake him ? 5. Two men are travelling the same way ; one travels at the rate of 38 miles a day, the other, who is behind the former, travels 44 miles ; how much does he gain of the first each day ? and in how many days would he gain 60 miles ? 1 6. A fox is 80 rods before a greyhound, and is running at the rate of 27 rods in a minute ; the grey- 82 ARITHMETIC. [Par/ 1, hound is following at the rate of 31 rods in a minute ^ in how many minutes will the greyhound overtake the fox. 7. If 2 yards of cloth cost 6 dollars, what would 4 yards cost? what would 12 yards cost ? 8. If 8 sheep cost 24 dollars, what would 3 cost ? i). if 4 tons of hay will keep 3 horses through the winter, how many tons will keep 30 horses the same time ? 10. If a man spends 8 shillings a day, how many dollars will he spend in a week ? 11. Bought ten pieces of cloth, each containing 5 yards, for 100 dollars ; what was it a piece ? and how much a yard ? 12. If the wages of 12 weeks come to 60 dollars, what is that a month ? and how much for 5 weeks. 13. If 7 horses eat 14 bushels of oats in 1 week, how many bushels would 15 horses eat in the same time ? 14. If 3 horses eat 8 bushels of oats in 2 weeks, how long would it take them to eat 40 bushels ? 15. If 1 horse eat 1 bushel of oats in 3 days, in how many days would 4 horses eat 36 bushels ? 16. If 2 men spend 12 dollars in 1 week, how many dollars would 3 men, at the same rate, spend in 5 weeks ? 17. If a staff 3 feet long cast a shadow of 2 feet Ht 12 o'clock, what is the length of a pole that casts a shadow 18 feet at the same time of day ? 18. If 47 gallons of water, in I hour, run into a cistern containing 108 gallons, and by a pipe 38 gal- lons run out in an hour, how much remains in the cistern in an hour ? and in how many hours will the cistern be filled ? 19. If 4 men can do a piece of work in 8 days, how many men would it take to do the same work in 4 days ? Sect. 6.] ARITHMETIC. 83 20. If 6 men can do a piece of worfc in 9 days, in how many days would 2 men do it ? 21. If 8 men can do a piece of work in 5 days, in how many days would they do a piece of work 4 times as large ? 22. If 7 men can do a piece of work in 3 days, how many men would it take to do a piece of work 5 times as large in the same time ? 23. If 8 men can do a piece of work in 4 days, in how many days would 2 men do a piece of work one half as large ? 24. A man bought a cask of wine containing 63 gallons, 3 sevenths of which leaked out ; and he sold the remainder for 36 dollars ; how much per gallon did he sell it for ? 25. If a cask of wine cost 48 dollars, what is 5 eighths of it worth ? 26. A man bought 7 oranges for 6 cents apiece, and sold them all for 54 cents, how much did he gain by the bargain ? 27. A man bought 8 yards of cloth for 56 dollars, and sold it again for 9 dollars a yard ; how much did he gain by the bargain ? 28. A man bought 8 barrels of flour for 40 dollars ; how much must he sell it at per barrel to gain 16 dollars ? 29. A man bought 5 firkins of butter at 7 dollars a firkin ; how much must he sell it per firkin to gain 10 dollars ? 30. A man gave 35 cents for his breakfast, which was 5 eighths of what he gave for his dinner ; what did he give for his dinner ? 31. A ship's crew of 6 men have provision for 3 months ; how many months would it last 1 man ? 32. A ship's crew have provision sufficient to last 1 oian 27 months ; how long would it last 9 men ? 33. A ship's crew have provision sufficient to 84 ARITHMETIC. [Par/ 1. last 3 men 10 months ; how long would it last 5 men ? 34. A man built 40 rods of wall in a certain time, another man can build 9 rods while the first builds 5 ; how much would he build in the same time ? SECTION VIL If the combinations in this section should be found too difficult, they may be omitted until reviewing the l>ook. A. 1. A MAN being asked the age of his eldest son, answered, that his youngest son was six years old, and that 2 thirds of the youngest son's age was just 1 fifth of the eldest son's age. Required the age of the eldest son. 2. 2 thirds of 6 is 1 fifth of what number ? 3. A man, being asked how many sheep he had, said that he had them in two pastures ; in one pasture he had eight ; and that 3 fourths of these was just 1 third of what he had in the other. How many were there in the other ? 4. 3 fourths of 8 is 1 third of what number ? 5. Two boys talking of their ages, one said he was 9 years old. Well, said the other, 2 thirds of your age is exactly 3 fourths of my age ; now, if you will tell me how old I am, I will give you as many apples as I am years old. What was his age ? 6. 2 thirds of 9 is 3 fourths of what number ? 7. Two boys counting their money, one said he had ten cents. The other says, 4 fifths of your money is exactly 2 sevenths of mine ; now, if you will tell how many I have, I will give you 1 half of them. How many had he ? 8. 4 fifths of 10 is 2 sevenths of what number r Sect 7.] ARITHMETIC. 85 9. 5 sixths of 12 is 2 thirds of what number ? 10. 6 sevenths of 14 is 4 ninths of what number? 11. '6 filths of 15 is 2 thirds of what number? 12. 7 ninths of 18 is 2 fifths of what number ? 13. 4 fifths of 20 is 8 sevenths of what number ? 14. 8 sevenths of 21 is 6 tenths of what number? 15. 5 sixths of 24 is 10 sevenths of how many times 5 ? 16. 3 sevenths of 28 is 2 eighths of how many times 7? 17. 4 fifths of 30 is 6 sevenths of how many times 8? 18. 6 eighths of 32 is 8 ninths of how many times 5? 19. 4 ninths of 36 is 8 tenths of how many times 6? 20. 3 fourths of 40 is 5 sevenths of how many times 8 ? 21. 6 ninths of 45 is 3 fifths of how- many times 7? 22. 5 sixths of 48 is 10 sevenths of how many times 3? 23. 4 sevenths of 63 is 6 fifths of how many times 8? 24. 5 ninths of 72 is 4 sevenths of how many times 9? B. 1. 4 fifths of 15 is 6 tenths of how many thirds of 21? 2. 4 thirds of 18 is 8 ninths of how many sevenths of 35 ? 3. 6 sevenths of 21 is 2 thirds of how many thirds of 24 ? 4. 5 fourths of 24 is 10 sevenths of how many fifths of 40. 5. 5 eighths of 32 is 2 fifths of how many fifths tf 35? 86 ARITHMETIC [Part \ 6. 4 sevenths of 63 is 6 eighths of how many ninths of 45 ? 7. 3 sevenths of 56 is 4 ninths of how many fourths of 28 ? 8. 3 eighths of 64 is 6 tenths of how many sixths of 30. 9. 2 eighths of 72 is 3 tenths of how many til'tha of 40? C, 1. Two times eleven are how many ? 2. Two times twelve are how many ? 3. Two times thirteen are how many ? 4. Two times fourteen are how many ? 5. Two times fifteen are how many ? 6. Two times sixteen are how many ? 7. Two times seventeen are how many ? 8. Two times eighteen are how many ? 9. Two times nineteen are how many ? 10. Two times twenty are how many ? 11. Three times eleven are how many ? 12. Three times twelve are how many ? 13. Three times thirteen are how many ? 1 4. Three times fourteen are how many ? 15. Three times fifteen are how many ? 16. Three times sixteen are how many ? 17. Three times seventeen are how many ? 1 8. Three times eighteen are how many ? 19. Three times nineteen are how many ? 20. Three times twenty are how many ? 21. Four times eleven are how many ? 22. Four times twelve are how many ? 23. Four times tliirteen are how many ? 24. Four times fourteen are how many ? 25. Four times fifteen are how many ? 26. Four times sixteen are how many? 27. Four times seventeen are how many 28. Four times eighteen are how many ? Sect. 7 ] ARITHMETIC. 87 29. Four times nineteen are how many '< 30. Four times twenty are how many ? 31. Five times eleven are how many ? 32. Five times twelve are how many ? 33. Five times thirteen are how many ? 34. Five times fourteen are how many ? 35. Five times fifteen are how many ? 36 Five times sixteen are how many ? 37. Five times seventeen are how many ? 38. Five times eighteen are how many ? 39. Five times nineteen are how many : 40. Five times twenty are how many ? 41. Six tinges eleven are how many ? 42. Six times twelve are how many ? 43. Six times thirteen are how many ? 44. Six times fourteen are how many ? 45. Six times fifteen are how many ? 46. Six times sixteen are how many ? 47. Six times seventeen are how many ? 48. Six tunes eighteen are how many ? 49. Six times nineteen are how many ? 50. Six times twenty are how many 'i 51. Seven times eleven are how many ? 52. Seven times twelve are how many ? 53. Seven times thirteen are how many ? 54. Seven times fourteen are how many ? 55. Seven times fifteen are how many ? 56. Seven times sixteen are how many ? 57. Seven times seventeen are how many ? 58. Seven times eighteen are how many ? 59. Seven times nineteen are how many ? 60. Seven times twenty are how many ? 61. Eight times eleven are how many ? 62. Eight times twelve are how many ? 63. Eight times thirteen are how many ? 64. Eight times fourteen are how many ? 65. Eight times fifteen are how many ? 88 ARITHMETIC. [Part ) 66. Eight times sixteen are how many ? 67. Eight times seventeen are how many ? 68. Eight times eighteen are how many ? 69. Eight times nineteen are how many ? 70. Eight times twenty are how many v 71. Nine times eleven are how many ? 72. Nine times twelve are how many ? 73. Nine tunes thirteen are how many ? 74. Nine times fourteen are how many ? 75. Nine times fifteen are how many ? 76. Nine times sixteen are how many ? 77. Nine times seventeen are how many ? 78. Nine times eighteen are how many ? 79. Nine tunes nineteen are how in any ? 80. Nine times twenty are how many ? 81. Ten times eleven are how many ? 82. Ten times twelve are how many ? 83. Ten times thirteen are how many ? 84. Ten times fourteen are how many ? 85. Ten times fifteen are how many ? 86. Ten times sixteen are how many ? 87. Ten tunes seventeen are how many ? 88. Ten times eighteen are how many ? 89. Ten times nineteen are how many ? 90. Ten times twenty are how many ? SECTION VIU. A. 1. IF you cut an apple into two equal parts., what is one of those parts called ? * 2. How many halves of an apple will make the whole apple ? 3. If you cut an apple into 3 equal parts, what is * See Section III., article B, remark before questions 1 axid 17 Sect. 8 j ARITHMETIC 89 1 of those parts called ? what are 2 of the parts called ? 4. How many thirds of an apple will make the whole apple ? 5. If you cut an apple into 4 equal parts, what is I of those parts called? what are 2 of those parta called ? what are 3 of them called ? 6. How many fourths of an apple make the whole apple ? 7. If an apple be cut into 5 equal parts, what is 1 of the parts called ? what are 2 of the parts called ? what are 3 of the parts called ? what are 4 of the parts called ? 8. How many fifths of an apple make the whole apple ? 9. If an apple be cut into 6 equal parts, what is 1 of the parts called ? what are 2 of the parts called ? what 3 ? what 4 ? what 5 ? 10. How many sixths of an apple make the whole apple ? 11. If an apple be cut into 7 equal parts, what is 1 of the parts called ? what are 2 of the parts called ? what 3 ? what 4 ? what 5 ? what 6 ? Let the instructor ask the pupils the divisions of a unit in this manner as far as the division into 10 parts It would be well to ask them further. Then let him begin again, and suppose an orange instead of an ap- ple. After applying the division to several different thing Si a line may be drawn on the blackboard and divided. It is well often to require the pupil to make the requisite division of a line on the board. 12. A man had a bushel of corn, and wished to give 1 half of a bushel apiece to some laborers ; bow many could he give it to ? 13. How many halves are there in 1 ? 90 ARITHMETIC. [/'art 1 14. A man divided 2 barrels of flour among his la- borers, giving them 1 half of a barrel apiece ; how many men did he give it to ? 15. How many halves are there in 2 ? * 16. In 3 bushels of corn how many half bushels ? 17. How many halves are there in 3 ? 18. A boy divided 4 oranges among his companions, giving them 1 half of an orange apiece ; how many boys did he give them to ? 19. How many halves are there in 4 ? 20. A man having some laborers, gave them 1 half a dollar apiece ; it took 3 dollars and 1 half a dollar to pay them ; how many laborers were there ? 21. How many halves are there in 3 and 1 half? 22. How many halves are there in 5 ? 23. How many halves are there in 7 and 1 half? 24. How can you tell how many halves there are in any number ? Answer. Since there are 2 halves in one, there must be twice as many halves as there are whole ones. 25. If you had 1 orange, and should divide it among your companions, giving them 1 third apiece, how many could you give it to ? 26. How many thirds are there in 1 ? 27. If you cut 2 oranges each hi to 3 pieces, how many pieces would they make ? 28. If you cut 3 oranges into 3 pieces each, how many pieces would they make ? 29. If you cut 4 apples each into 3 pieces, how many pieces would they make ? 30. How many thirds are there in2?in3?in4? in5? 31. How can you tell how many thirds there are io any number ? Answer. Since there are 3 thirds in one, there will be 3 times as many thirds as there are whole ones ? * See tfao Key Sect. 8.] ARITHMETIC. 9} 32. If you had 2 bushels and 1 third of a bushel of corn to give to some poor persons, how many could you give it to if you should give them 1 third of a bushel apiece ? 33. How many thirds are there in 2 and 1 third ? 34. If a horse can eat 1 third of a bushel of oats in 1 day, how many days would it take him to eat 3 bushels and 2 thirds of a bushel ? 35. How many thirds are there in 3 and 2 thirds ? 36. If 1 horse can eat 1 third of a bushel of oats in a day, how many horses will it take to eat 5 bushels and 2 thirds of a bushel in the same time ? 37. In 5 and 2 thirds how many thirds ? 38. In 7 and 1 third how many thirds ? 39. If 1 horse will eat 1 fourth of a ton of hay in 1 month, how many horses will eat a ton in the same time ? How many will eat 2 tons ? How many will eat 3 tons ? 4 tons ? 5 tons ? 6 tons ? 40. How many fourths are there inl? in2? in3? in 4 ? in 5 ? in 6 ? 41. How can you tell how many fourths there are in any number ? 42. How many fourths are there in 4 and 1 fourth ? 43. In 5 tons of hay and 3 fourths of a ton how many fourths of a ton ? 44. In 5 and 3 fourths how many fourths ? 45. In 7 and 1 fourth how many fourths ? 46. In 9 **nd 3 fourths how many fourths ? 47. If a horse eat 1 fifth of a ton of hay in a month, how many horses will eat a ton in the same time ? 2 tons ? 3 tons ? 4 tons ? 5 tons ? 48. How many fifths are there in 1? in2? in 3? in 4 ? in 5 ? in 7 ? in 9 ? 49. How can you tell how many fifths there are in any number ? 50. In 2 and 1 fifth how many fifths ? 51. In 3 dollars and 2 fifths of a dollar how many fifths of a dollar ? 92 ARITHMETIC. [Part 1 52. Jn 3 and 2 fifths how many fifths ? 53. In 5 and 3 fifths how many fifths ? 54. In 6 and 4 fifths how many fifths ? 55. How many sixths are there in 1 ? in 2 ? in 3 ? in 4 ? in 5 ? in 7 ? in 8 ? 56. In 2 and 2 sixths how many sixths ? 57. In 3 and 4 sixths how many sixths ? 58. In 6 and 5 sixths how many sixths ? 59. How many sevenths are there in 1 ? in 2 ? in 3? in 4? in 6? "in 9? in 10? 60. In 3 and 1 seventh how many sevenths? 61. In 5 and 3 sevenths how many sevenths? 62. In 7 and 5 sevenths how many sevenths ? 63. How many eighths are there in 1 ? in 3 ? in 5 ? in 8? 64. In 2 and 3 eighths how many eighths ? 65. In 3 and 5 eighths how many eighths ? 66. In 5 and 7 eighths how many eighths ? 67. How many ninths are there inl?in2?in7? in 5? 68. In 2 and 2 ninths how many ninths ? 69. In 4 and 3 ninths how many ninths ? 70. In 6 and 4 ninths how many ninths ? 71. In 8 and 7 ninths how many ninths ? 72. How many tenths in 1 ? in 2 ? in 5 ? in 8 ? 73. In 3 and 3 tenths how many tenths ? 74. In 4 and 7 tenths how many tenths ? 75. In 8 and 9 tentlis how many tenths ? 76. In 7 and 4 tenths how many tenths ? 77. In 9 and 8 tenths how many tenths ? 78. In 7 and 4 sevenths how many sevenths ? 79. In 9 and 2 thirds how many thirds ? 80. In 10 and 3 fourths how many fourths ? 81. In 8 and 4 fifths how many fifths ? 82. In 7 and 5 ninths how many ninths ? B. 1. If you give 4 men 1 half of a barrel of floui apiece, how many barrels will it take ? Sect. 8 ] ARITHMETIC. 93 2. In 4 halves how many times 1 ? 3. If you give 3 boys 1 half of an orange apiece, how many oranges will it take ? t. In 3 halves how many times 1 ? . If you give five men 1 half of a dollar apiece, how many dollars will it take ? 6. In 5 halves how many times 1 ? 7. In 6 halves how many times 1 ? 8. In 7 halves how many times 1 ? 9. How can you tell how many whole ones there are in any number of halves ? 10. A man divided some corn among 6 persons, giving them 1 third of a bushel apiece ; how many bushels did it take ? 11. In 6 thirds how many times 1 ? 12. In 5 thirds how many times 1 ? 13. A man gave eight paupers 1 third of a dollar apiece, how many dollars did it take ? 14. In 8 thirds how many times 1 ? 15. In 10 thirds how many times 1 ? 1 6. How can you tell how many whole ones there are in any number of thirds ? 17. If a man spends 1 fourth of a dollar in one day, how many dollars will he spend in 8 days ? How many in 7 days ? How many in 11 days ? 18. In 8 fourths how many times 1 ? 19. In 7 fourths how many times 1 ? 20. In 1 1 fourths how many times 1 ? 21. In 13 fourths how many times 1 ? 22. In 18 fourths how many times 1 ? 23. How can you tell how many whole ones there are in any number of fourths ? 24. If 1 fifth of a barrel of flour will last a family 1 day, how many barrels will last them 10 days ? How many 8 days ? 11 days ? 17 days ? 25. In 10 fifths how many times 1 ? 26. In 8 fifths how many times 1 ? 94 ARITHMETIC. [Part I 27. In 1 1 fifths how many times 1 ? 28. In 17 fifths how many times 1 ? 29. In 18 sixths how many times 1 ? 80. In *2o fifths how many times 1 ? 31. In 21 sevenths how many times 1 ? 32. In 24 eighths how many times 1 ? 33. In 36 ninths how many times 1 ? 34. In 30 tenths how many times 1 ? 35. In 35 fourths how many times 1 ? 36. In 37 eighths how many times 1 ? 37. In 43 fifths how many times 1 ? 38. In 48 ninths how many times 1 ? 39. In 53 tenths how many times 1 ? 40. In 57 eighths how many times 1 ? 41. In 76 tenths how many times 1 ? 42. In 78 ninths how many times 1 ? SECTION IX. A. 1. IF a hreakfast for 1 man cost 1 third of a dollar, what would a breakfast for 2 men cost ? 2. How much is 2 times 1 third ? 3. If it take you 1 third of an hour to travel 1 mile, how long will it take you to travel 3 miles ? 4. 'How much is 3 times 1 third ? 5. If 1 man can eat 1 third of a pound of meat at a meal, how much can 5 men eat ? 6. How much is 7 times 1 third ? 7. If 1 man can eat 2 thirds of a pound of meat for dinner, how many thirds of a pound would 3 men eat ? 8. How much is 2 times 2 thirds ? 9. A man gave to 4 paupers 2 thirds of a dollar ipiece, how many thirds of a dollar did he give them ? how many dollars ? 10. 5 times 2 thirds are how many thirds? how many times 1 ? Sect. 9.] ARITHMETIC. 95 1 1 . Tf you gh e 3 men 1 fourth of a dollar apiece, how many fourths of a dollar will it take ? 1 2. 3 times 1 fourth are how many fourths ? 13. If you give 3 men 3 fourths of a bushel of corn apiece, how many fourths of a bushel will it take ? how many bushels ? 14. 5 times 3 fourths are how many fourths? how many times 1 ? 15. If 1 horse eat 1 fifth of a bushel of oats in a day, how much will 4 horses eat in the same time ? 16. 3 times 1 fifth are how many fifths ? 17. If 1 man can earn 3 fifths of a dollar in a day, how much can he earn in 4 days ? 18. 7 times 3 fifths are how many fifths ? how many times 1 ? 19. If a family consume 2 sevenths of a barrel of dour in a week, how much would they consume in 5 weeks ? 20. 6 times 2 sevenths are how many sevenths ? how many times 1 ? 21. 5 times 3 eighths are how many eighths ? low many times 1 ? 22. How much is 6 times 3 fifths ? 23. How much is 7 times 5 sixths ? 24. How much is 5 times 4 ninths ? 25. How much is 6 times 8 ninths ? 26. How much is 7 times 9 tenths ? 27. How much is 5 times 7 tenths ? 28. How much is 6 times 7 eighths ? 29. How much is 9 times 5 eighths ? 30. How much is 8 times 5 sevenths ? 31. How much is 7 times 5 sixths ? 32. How much is 8 times 7 fourths ? 33. How much is 7 times 4 fifths ? 34. How much is 5 times 3 eighths ? B. 1. If 1 bushel of wheat cost a dolla, and 1 half what will 2 bushels cast ? 96 ARITHMETIC. [Part I 2. How much is 2 times 1 and 1 half* * 3. If a barrel of cider cost 2 dollars and a half, what will 3 barrels cost ? 4. How much is 4 times 2 and 1 half? 5. If a barrel of beer cost 3 dollars and a half, irhat will 2 barrels cost ? 6. How much is 5 times 3 and 1 half? 7. How much is 6 times 3 and 1 half? 8. If a box of butter cost 2 dollars and 1 third of a dollar, what will 3 boxes cost ? 9. How much is 4 times 2 and 1 third ? 10. If you give to two persons 3 bushels and a third of a bushel of wheat apiece, how many bushels will it take ? 11. How much is 5 times 3 and 1 third? 12. If you give to 4 persons each 2 oranges and 1 fourth of an orange, how many oranges will it take? 13. How much is 5 times 2 and I fourth ? 14. If it take 3 yards and 2 thirds of a yard of cloth to make a suit of clothes, how many yards will it take to make 2 suits ? 15. How much is 4 times 3 and 2 thirds ? 16. If a family consume 2 bushels and 2 thirds of a bushel of malt in 1 month, how much will they con- sume in 3 months ? 17. How much is 5 times 2 and 2 thirds ? 18. How much is 4 times 3 and 3 fourths ? 19. How much is 2 times 3 and I fourth ? 20. How much is 3 times 3 and 3 fourths ? 21. How much is 3 times 5 and I fourth ? 22. If a horse eat 3 tons and 1 fifth of a ton of hay in a year, how much will 2 horses eat in the same time ? 23. How much is 4 times 3 and 1 fifth. ? * This is to be understood 2 times 1 and 2 times 1 half, and tc be answered thus: 2 times 1 are 2, and 2 times 1 half are 2 halves or 1, which, added to 2, rm.kes 3. Sect. 9.J ARITHMETIC. 97 24. If a man can travel 4 miles and 2 fifths of a mile in one hour, how far will he travel in 3 hours ? 25. How much is 5 times 4 and 2 fifths ? 26. How much is 3 times 5 and fifths ? 27. How much is 4 times 6 and 2 fifths ? 28. How much is 3 times 2 and 1 sixth ? 29. How much is 2 times 5 and 3 sixths ? 30. If a yard of cloth cost 4 dollars and 5 sixths of a dollar, what will 4 yards cost ? 31. How much is 7 times 4 and 5 sixths? 32. How much is 2 times 3 and 3 sevenths ? 33. How much is 3 times 4 and 3 sevenths ? 34. If a bushel of wheat cost 7 shillings and 8 eighths, what will 5 bushels cost at that rate ? * 35. How much is 3 times 4 and 5 eighths ? 36- How much is 4 times 3 and 7 eighths f 37. A man bought 8 yards of cloth, at 9 dollars and 3 tenths a yard ; how much did it come to ? 38. How much is 6 times 2 and 2 ninths ? 39. How much is 4 times 5 and 3 ninths ? 40. A man bought 10 barrels of cider at 3 dollars and 5 sixths a barrel ; how much did it come to? 41. How much is 3 times 7 and 2 tenths ? 42. What cost 8 barrels of beef at 9 dollars and 3 sevenths a barrel. C. 1. A boy wished to give 5 other boys 1 half pint of chestnuts apiece ; how many pints would it take ? 2. A boy wished to give 3 other boys 3 fourths of an orange apiece ; how many oranges would it take ? 3. A man gave to 10 persons 1 fifth of a bushel of eorn apiece ; how many bushels did it take ? 4. A man gave to 7 men 3 fourths of a gallon of beer apiece ; how many gallons of beer did it take ? 5. If it take 1 yard and 1 fourth of a yard of cloth to make a pair of pantaloons, how many yards would it take to make 8 pairs ? 98 ARITHMETIC. [Part I. 6. If a family consume 2 bushels and 3 fifths of a bushel of grain in 1 week, how many bushels would they consume in 4 weeks? 7. If a horse eat 3 bushels and 4 sevenths of a bushel of oats in 1 week, how many bushels would he eat in 8 weeks? 8. If a horse eat 5 loads and 3 eighths of a load of hay in 1 year, how many loads would 6 horses eat? 9. If a man travel 4 miles and 5. ninths in an hour, how many, miles would he travel in 8 hours ? 10. If in an orchard of 10 trees, each tree bears 8 bushels and 3 sevenths, how many bushels will the whole orchard bear? 11. If a man can build 5 rods and 7 eighths of a rod of wall in 1 day, how many rods can he build in 8 days ? 12. If 3 men can build a piece of wall in 4 days and 3 fifths of a day, how many days would it take 1 man to build it ? 13. If 1 man can build 7 rods and 2 fifths of wall in a day, how many rods would 1 men build ? ' 14. If 1 man can build 3 rods and 2 ninths of wall in 1 day, how many rods would 3 men build in 4 days ? 15. If it take 1 yard and 3 sevenths of a yard of cloth to make 1 pair of pantaloons, and 2 yards and 4 sevenths for a coat ; how many yards would it take to make 3 pairs of pantaloons and 3 coats. SECTION X. A. 1. IF a yard of cloth cost 3 dollars, what wil) 1 half of a yard cost ? 2. What is 1 half of 3 ? * * See Section V.. article A Sect. 10.] ARITHMETIC. 99 3. If a barrel of beer cost 5 dollars, what will 1 half of a barrel cost ? 4. What is 1 half of 5 ? 5. If 2 barrels of cider cost 7 dollars, what is that a barrel ? 6. What is 1 half of 7 ? 7. What is 1 half of 9 ? 8. What is 1 half of 11 ? 9. What is 1 half of 13 ? 10. What is 1 half of 15 ? 11. If you divide 1 bushel of wheat equally among 3 persons, what part of a bushel will you give them apiece ? 12. If .3 yards of cloth cost 1 dollar, what part of a dollar will 1 yard cost ? 13. What is 1 third of 1 ? 14. How could you divide 2 oranges into 3 equal parts ? that is, how can you find 1 third of 2 or- anges ? * 15. One third of 2 oranges will be the same as how many thirds of 1 orange ? 16. If you divide 2 bushels of wheat equally among 3 persons, what part of a bushel will you give them apiece ? 17. If 3 bushels of corn cost 2 dollars, what part of a dollar will 1 bushel cost ? NOTE. One third of two things is twice as much as one third of one thing. One third of one is one third, and consequently one third of two things is two thirds. In the same manner, one third of four things is four thirds of one thing. If four oranges be cut each into three parts, and then one part of each be taken, it will make four pieces, each of which is one third of one * Divide each orange into three parts, and then take one part from each. 100 AU1THMET1C. [Part 1 orange. Hence one third of four oranges is four third* of one orange, that is, one whole one and one third. 18. If 3 bushels of wheat cost 4 dollars, how much is that a bushel ? 19. What is one third of 2 ? of 4 ? 20. If 3 gallons of wine cost 5 dollars, what is that a gallon ? 21. What is 1 third of 5 ? of 7 ? of 8 ? of 10 ? of 11. 22. If a bushel of apples be divided equally among 4 persons, what part of a bushel will they have apiece ? What would they have apiece, if 2 bushels were di- vided among them ? What if 3 bushels ? What if 5 bushels ? What if 6 bushels ? 23. What is 1 fourth of 1 ? of 2 ? of 3 ? of 5 ? of 6 ? of 7 ? of 9 ? of 10 ? 24. If a bushel of malt will serve 5 persons 1 month, how much will serve 1 person the same time ? 25. If 2 barrels of cider will serve 5 persons 1 month, how much will serve 1 person the same time. 26. If 3 barrels of flour be divided among 5 men, how much will each have ? If 4 barrels were divided, what would each have ? What if 6 barrels were divided ? What if 7 barrels were divided ? 27. What is 1 fifth of 1 ? of 2 ? of 3 ? of 4 ? of 6 ? of 7 ? 28. What is 1 sixth of 1 ? of 2 ? of 3 ? of 4 ? of 5 ? of 7 ? of 8 ? of 9 ? of 10 ? 29. What is 1 seventh of 1 ? of 2 ? of 3 ? of 4 ? of 5 ? of 6 ? 30. What is 1 eighth of 1 ? of 2 ? of 3 ? of 4 ? of 5 ? of 6 ? of 7 ? of 8 ? of 9 ?. of 10 ? 31. What is 1 ninth of 1 ? of 2 ? of 3 ? of 4 ? of 5 ? of 6 ? of 7 ? of 8 ? of 9 ? of 10 ? of 11 ? 32. What is 1 tenth of 1 ? of 2 ? of 3 ? of 4 ? of 5 ? of 6 ? of 7 ? of 8 ? of 9 ? of 10 ? of 11 ? of 12? of 13? Sect. 10.] ARITHMETIC. 101 33. If 3 yards of cloth cost 2 dollars, what will 1 yard cost ? What will 2 yards cost ? I ; ,. > 34. If 1 bushel of wheat cost 2 dollpir.VVhaVwill 1 third of a bushel cost ? What will 2 thirds of a bushel cost ? 35. What is 1 third of 2 ? 36. What is 2 thirds of 2 ? 37. If a load of wood cost 5 dollars, what will 1 third of a load cost ? What will 2 thirds of a load cost? 38. What is 1 third of 5 ? 39. What is 2 thirds of five ? 40. What is 2 thirds of 7 ? 41. If 4 bushels of salt cost 3 dollars, what will 1 bushel cost ? WTiat will 3 bushels cost ? 42. What is 1 fourth of 3 ? 43. What is 3 fourths of 3 ? 44. If a barrel of cider cost 2 dollars, what will 1 fifth of a barrel cost ? What will 2 fifths of a barrel cost? 45. What is 1 fifth of two ? 46. What is 2 fifths of 2 ? 47. What is 1 fifth of 7 ? 48. What is 3 fifths of 7 ? 49. If 7 gallons of wine cost" 5 dollars, what will a gallon cost ? What will 4 gallons cost ? 50. What is 1 seventh of 5 ? 51. What is 4 sevenths of 5 ? 52. What is 1 sixth of 4 ? 53. What is 5 sixths of 4 ? 54. If you divide 7 dollars among 8 men, what part of a dollar will you give them apiece ? What would three of them have ? 55. What is 1 eighth of 7 ? 56. What is 3 eighths of 7 ? 57. What is 1 ninth of 10 ? 58. What is 7 ninths of 10 ? 102 ARITHMETIC. [Part i. 59. What is 1 tenth of 14 ? 60. What is '4 tenths of 14 ? 61. If 5 yar of what number ? Miscellaneous Examples. 1. A man sold 8 yards of cloth for 3f dollars a yard ; what did it come to ? 2. A man sold a horse for 76 dollars, winch was of what it cost him ; how much did it cost him ? 3. A man sold of a gallon of wine for 40 cents ; what was that a gallon ? 4. If it will take 1J yards of cloth to make a coat, how many yards will it take to make 7 coats ? 5. If 1 horse consume 3f bushels ~of oats in 2 days, how much would 2 horses consume in 5 days ? 6. If, when the days are 9J hours long, a man per- form a journey in 10 days, in how many days would he perform it when the days are 12 hours long ? 7. A man sold 8 yards of cloth for 7$ dollars a yard, and received 8 firkins of butter at 6f dollars a firkin ; how much was then due to him ? 8. Two men are 38 miles apart, and are travelling awards each other, one at the rate of 3 miles an hour, the other 2 miles ; how much do they approach each other in an hour ? How much in 2 hours ? In how many hours will they meet ? At what distance from each place from which they set out ? Sect. 1 3.] ARITHMETIC. 1 ] 5 SECTION XIII. A. 1. If you give J of an orange to one boy, and J to another, how much more do you give the first than the second ? 2. J of an orange is how many J of an orange ? 3. If you give J of an orange to one boy, and to another, how many ^ would you give away ? How many J would you have left ? 4. J and J are how many ? 5. A man gave to one laborer J of a bushel of wheat, and f to another ; how many of a bushel did he give to both ? How many bushels ? 6. J and f are how many ? How many times 1 ? 7. A man gave ^ of a barrel of flour to one man, and J of a barrel to another ; to which did he give the most ? 8. J is how many J ? 9. A man bought ^ of a bushel of wheat at one time, and f of a bushel at another ; at which time did he buy the most? 10. ^ is how many J ? 11. A man bought of a yard of cloth at one time, and f of a yard at another ; at which time did he buy the most ? 12. are how many J ? 13. A man wished to give J of a bushel of wheat to one man, and J of a bushel to another ; but he could not tell how to divide it. Another man standing by advised him to divide the whole bushel into six equal parts first, and then take J of them for one, and ^ of them for the other. How many parts did he give to each ? How many to both ? How many had he left ? 14. J is how many J ? J is how many J ? \ and \ are how many \ ? 15. A man paying some money to his laborers, 1 1 6 ARITHMETIC. [Part 1 gave each man J of a dollar, and each boy of a dol- lar ; how much more did he give to a man than to a boy? 16. What is the difference between and J ? 17. If a man can earn J of a dollar in a day, and a boy of a dollar, how much does the man earn more than the boy ? 18. What is the difference between j and ? 19. A boy distributing some nuts among his com- panions, gave J of a quart to one, and J of a quart to another ; how much more did he give to one, than to the other ? Note. Change them to sixths. 20. What is the difference between and J ? 21. A man having two bushels of grain to distrib- ute among his laborers, wished to give J of a bushel to one, and f of a bushel to another, and the rest to a third ; but was at a loss to tell how to divide it ; at last he concluded to divide each bushel into six equal parts, or sixths, and then to distribute those parts. How many sixths did he give to each ? 22. is how many J ? 23. A man had a horse, and a cow, and a sheep. The horse would eat of a load of hay in the winter, the cow J, and the sheep J. How many J of a load would each eat ? How many J would they all eat ? How many loads ? 24. A boy having a quart of nuts, wished to divide them, so as to give one companion J, another J, and a third \ of them ; but in order to make a proper divis- ion, he first divided the whole into eif^t equal parts, and then he was able to divide them as he wished. How many eighths did he give to each ? How many eighths had he left for himself? 25. J is how many \ ? % is how many ? and J and | are how many ? 26. A man gave J of a barrel of flour to one man, Sect. 13.] ARITHMETIC. 117 and f of a barrel to another ; to which did he give the most ? How much ? 27. Which is the larger, f or ? How much the larger ? 28. A boy, having a pound of almonds, said 'he in- tended giving J of them to his sister, and J to his broth- er, and the rest to his mamma. His mamma smiling said she did not think he could divide them so. Oh, yes, I can, said he, I will first divide them into twelve equal parts, and then I can divide them well enough. Pray how many twelfths did he give to each ? 29. J is how many ^ ? is how many ^ ? J and J are how many -^ ? 30. Mr. Goodman having a pound of raisins, said he would give Sarah J, and Mary ^, and James of them, and he told Charles he should have the rest, if he could tell how to divide them. Well, said Charles, I would first divide the whole into twelve equal parts, and then I could take ^ and and J of them. How many twelfths would each have ? 31. J and and J are how many ^ ? 32. George bought a pine-apple, and said he would give | of it to his papa, and f to his mamma," and y 3 ^ to his brother James, if he could divide it. James took it, and cut it into twenty equal pieces, and then distrib- uted them as George had desired. How many twen- tieths did he give to each ? 33. is how many ^ ? is how many ^ ? is how many ^ ? T ^ i s how many ^ ? 34. J is how many T ij ? 35. J is how many ^ ? 36. ^ is how many \ ? 37. is. how many ^ ? 38. are how many 39. are how many 40. is how many - 41. f are how many 118 ARITHMETIC. [Parti. 42. are how many T ^ ? 43. f are how m&ny ^ ? 44. are how many ^ ? 45. | are how many -fa ? 46. f are how many ^ ? 47. ^y are how many ^ ? 48. Reduce to sixths and to sixths. 49. and are how many ? 50. Reduce and to eighths. 51. and are how many | ? 52. J and are how many ? 53. and are how many ? 54. and f are how many J ? 55. and f are how many ? 56. and are how many T \j ? 57. and and | are how many | ? 58. J and f and T ^ are how many ^ ? 59. and are how many ^ ? 60. and I and are how many ^ ? 61. and ^ and are how many ^ ? 62. and and f and J and T ^ are how many^ ? 63. ^ and ^ are how many T ^ ? 64. and ^ are how many T ^ ? 65. and are how many -^ ? 66. less are how many | ? 67. f and f , less ^, are how many T \j ? 68. less f are how many ^ ? 69. less f are how many ^ ? 70 less f are how many ^ ? 71 , and f , and , and ^, less f , are how many ^ ? 72. i, and , and f , and ^, and ^y, less |, are how many ^ ? 73. | and f are how many ^ ? 74. and f are how many 3^ ? 75. f and f are how many ^ ? When the denominators in two or more fractions are .Sect. 13.] ARITHMETIC. 119 the same, the fractions are said to have a common denominator. Thus f and f have" a common denomina- tor. We have seen that, when two or more fractions have a common denominator, they may be added and subtracted as well as whole numbers. We add or sub- tract the numerators, and write their sum or difference over the common denominator. The first part of the process in the above examples was to reduce them to a common denominator. 76. Reduce f and f to a common denominator. Note. They may be reduced to twelfths. If it cannot be immediately seen what number must be the common denominator, it may be found by mul- tiplying all the denominators together ; for that will always produce a number divisible by all the denomi- nators. 77. Reduce f and f to a common denominator. 78. Reduce and f and } to a common denominator. 79. Reduce- and f to a common denominator. 80. Reduce f and f to a common denominator. 81. Reduce |- and f and f to a common denominator. 82. Add together f and f . 83. Add together f and T %. 84. Add together f and T %. 85. Add together J and f and fa 86. Subtract ^ from J. 87. Subtract ^ from f . 88. Subtract f from f . 89. Subtract f from . B. 1. Mr. F. said he would give of a pine-apple to Fanny, and f to George, and the rest to the one that could tell how to divide it, and how much there would be left. But neither of them could tell ; so he kept it viimself. Could you have told, if you had been there ? How would von divide it ? How much would be left ? 120 ARITHMETIC. [Part \ 2. A man sold 1 bushels of wheat to one man, 4vj bushels to another ; how many bushels did he sell to both? 3. A man bought 6 bushels of wheat at one time, and 2 at another ; how much did he buy in the whole ? 4. A man bought 7| yards of one kind of cloth, and 6J yards of another kind ; how many yards in the whole ? 5. A man bought f of a barrel of flour at one time, 2 barrels at another, and 6| at another ; how much did he buy in the whole ? 6. A man bought one sheep for 4$ dollars, and an- other for 5 dollars ; how much did he give for both ? 7. There is a pole standing, so that J of it is in the mud, and of it in the water, and the rest out of the water ; how much of it is out of the water ? 8. A man having undertaken to do a piece of work, did i of it the first day, J of it the second day, and of it the third day, how much of it did he do in three days ? 9. A man having a piece of work to do, hired two men and a boy to do it. The first man could do J of the work in a day, and the other of it, and the boy of it ; how much of it would they all do in a day ? Note. By dividing a line into halves, % and then into fourths, it will be seen that is the same as ; a line di- vided into halves and then into sixths, will show that is the same as , and 4 as f ; f , $, can therefore be reduced to -J, and | to f . This is called reducing fractions to their lowest terms. It is done by dividing by the greatest num- ber that will divide it without a remainder. C. 1. Reduce f to its lowest terms.* Am. f . 2, Reduce ^ to its lowest terms. 3. Reduce f to its lowest terms. * If this article should be found too difficult for the pupil, he may Dmit it till after the next section. Sect. 14.] ARITHMETIC. 121 4. Reduce fa to its lowest terms. 5. Eeduce | to its lowest terms. 6. Reduce ^ to its lowest terms. 7. Reduce ^ to its lowest terms. 8. Reduce ^ to its lowest terms. 9. Reduce Jf to its lowest terms. 10. Reduce $ to its lowest terms. 11. Reduce f to its lowest terms. 12. Reduce J to its lowest terms. 13. Reduce f to its lowest terms. 14. Reduce f \ to its lowest terms. Note. It will be seen by the above section that ii both the numerator and denominator be multiplied by the same number, the value of the fraction will not be altered ; or if they can both be divided by the same number without a remainder, the fraction will not be altered. SECTION XIV. A. 1. A BOY having J of an orange gave away ^ of that, what part of the whole orange did he give away ? 2. What is i of J ? 3. If you cut an apple into three pieces, and then cut each of those pieces into two pieces, how many pieces will the whole apple be cut into ? What part of the whole apple will one of the pieces be ? 4. What is J of J ? 5. A boy had J of a pine apple, and cut that half into three pieces, in order to give away ^ of it. What part of the whole apple did he give away ? 6. What is of ? 7. If an* orange be cut into 4 parts, and then each of the parts be cut in two, how many pieces will the whole be cut into ? 8. What is 4 of J ? 1 22 ARITHMETIC. [Part 1. 9. A man having J of a barrel of flour, sold | of that ; how much did he sejil ? 10. What is i of J? 11. If an orange be cut into 4 equal parts, and each of those parts be cut into 3 equal parts, how many parts will the whole orange be cut into ? 12. What is J of J ? 13. A boy having of a quart of chestnuts, gave away J of what he had. What part of the whole quart did he give away ? 14. What is of ? 15. What is J of J ? 1 6. A man owning J of a ship, sold ^ of his share : what part of the ship did he sell, and what part did he then own ? 17. What is J of J ? 18. What is J of i ? 19. What is J of J ? 20. What is J of ? 21. What is | of | ? 22. What is of ? 23. What is | of | ? 24. What is J of ? 25. What is | of | ? 26. What is of jf 27. What is of ? 28. What is } of \ ? 29. A boy having of an orange, (that is, 2 pieces,) gave his sister J of what he had ; how many thirds did he give her ? 30. What is J of ? 31. A boy having f of a pine apple, said he would give one half of what he had to his sister, if she could tell how to divide it. His sister says, you have got f , or three pieces, if you cut them all in two, you can give me J of them. But J of J is J, therefore i shall have f of the whole pine apple. Sect. 14.] ARITHMETIC. 123 32. What is J of | ? 33. A man owning f of a share in the Boston Bank, sold J of his part. What part of a share did he sell ? 34. What is J- of f ? 35. A man owning f of a ship, sold ^ of his share ; what part of the whole ship did he sell ? What part had he left ? 36. What is J of J ? 37. What is J of | ? 38. What is | of f ? 39, -What is | of f ? 40. What is J of V 2 - ? 41. What is J of ? ? 42. A man owning f of a share in the Boston Bank, sold J of his part ; what part of a whole share did he sell ? 43. What is J of f ? 44. What is J of f ? 45. A boy having of a water-melon, wished to divide his part equally between his sister, his brother, and himself, but was at a loss to know how to do it ; but his sister advised him to cut each of the fifths into 3 equal parts. How many pieces did each have ? and what part of the whole melon was each piece ? 46. What is J of ? 47. What is J of | ? 48. What is J of f ? 49. What is of f ? 50. What is ^y of | ? 51. What is J of J ? 52. What is of i ? 53. What is J of ? 54. What is | of | ? 55. What is of f ? 56. What is | of f ? 57. What is J of f ? 124 ARITHMETIC. [Part 1 58. What is of g ? 59. What is of | ? 60. What is \ of f ? 61. What is f of ? 62. What is | of f ? 63. What is f of f ? 64. What is ^ of } ? 65. What is ^ of } ? 66. What is | of ^ ? 67. What is f of ? 68. What is of f- ? 69. If a yard of cloth cost 2J dollars, what will \ of a yard cost ? 70. What is J of 2 J ? 71. A boy had 2 oranges, and wished to givjB J of them to his sister, and to his brother, but he did not know how to divide them equally. His brother told him to cut the whole into halves, and then cut each of the halves into 3 pieces. What part of a whole orange did each have ? 72. What is J of 2 J ? 73. A man bought 4 bushels of corn for 3 dollars ; what part of a dollar did 1 bushel cost ? Change the 3| to thirds, and then find J of y as above. 74. What is i of 5 ? 75. If 5 bushels of wheat cost 7J dollars, what is that a bushel ? 7. What is J- of 7f ? 77. A man bought 6 gallons of alcohol for 8f dol- lars ; what was that a gallon ? 78. What is J of 8 ? 79. A man bought 7 gallons of wine for 8f dollars how much was that a gallon ? 80. What is } of 8f ? 81. A man bought 10 pieces of nankin for 6| dol- lars ; how much was it a piece ? Sect. H.] ARITHMETIC. 125 82. What is ^ of 6| ? 83. If 9 bushels of rye cost 7f dollars, what is that a bushel ? 84 What is ef 7% ? 85. What is J of 5f ? 86. What is | of 8j ? 87. What is i of 6 A ? 88. What is of 9 ? 89. A man bought 7 yards of cloth for 18 J dollars ; what was that a yard ? What would 3 yards cost at that rate ? 90. What is | of 18 J ? What is f of 18 ? 91. A man bought 5 barrels of cider for 27 dol- lars ; what was it a barrel ? What would 7 barrels cost at that rate ? 92. What is j- of 27| ? What is J of 27f ? 93. If 6 barrels of flour cost 38f dollars, what would 10 barrels cost at that rate ? 94. What is V of 38$ ? B. 1. A man bought a piece of cloth for 42f dol- lars, and was obliged to sell it for of what it cost him ; how much did he lose ? 2. A man bought a quantity of flour for 53f dol- lars, and sold it for f of what it cost him ; how much did he gain ? . 3. If 7 men can do a piece of work in 4f days, how long will it take 1 man to do it ? How long will it take 3 men to do it? 4. If 4 men can do a piece of work in 9f days, how long would it take to do it 'if 7 men were em ployed ? 5. There is a pole standing so that f of it is in the water, and as much in the mud ; how much is in the mud ? 6. If a man can travel 13f miles in 3 hours, how many miles will he travel in 8 hours ? 126 ARITHMETIC. [Part 1. 7. If 5 horses will eat 26f loads of hay in a year, what will 8 horses eat in the same time ? 8. If. 4 pipes will empty a cistern in 6g hours, how long will it take 7 pipes of. the same size tc empty it?. SECTION XV. A. 1. A BOY having 2 oranges wished to give of an orange apiece to his playmates ; how many could he give them to ? If he had given of an orange apiece, how many could he have given them to ? 2. How many times are there in 2 ? How many times are there in 2 ? 3. A man having 3 bushels of corn, distributed it among some poor persons, giving them f of a bushel each ; to how many did he give it ? Note. First jind how many he would have given it to, if he had given of a bushel to each. 4. In 3 are how many times ^ ? How many times f ? 5. If f of a barrel of flour will last a family one month, how long will 4 barrels last the same family ? How long will 6 barrels last ? How long will 10 barrels last ? 6. How many times is contained in 4 ? How many times in 6 ? How many times in 10 ? 7. If f of a bushel of wheat will last a family one week, how many weeks will 6 j bushels last the same family ? 8. How many times is f contained in 6 J ? 9. There is a cistern having a pipe which will fill it in f of an hour ; how many tunes would the pipe fill the cistern in 3| hours ? 10. How many times is | contained in 3f ? 11. How much cloth at 1^ dollars (that is f dollars) a yard can be bought for 4 dollars ? 12. How many times is 1 J or | contained in 4 ? Sect. 15.] ARITHMETIC. 127 13. A man distributed 8^ bushels of wheat among Borne poor persons, giving 1 bushels to each ; how many did he give it to ? 14. How many times is 1^ contained in 8J ? 15. If a soldier is allowed 1 pounds (that is j of a pound) of meat in a day, to how many soldiers would 6 1 pounds be allowed ? 16. How many times is 1^ contained in 6? 17. If If tons of hay will keep a horse through the winter, how many horses will 10 tons keep ? 18. How many times is If contained in 10 ? 19. At 2^ dollars a box, how many boxes of raisins can be bought for 10 dollars ? 20. How many times is 2^ contained in 10 ? 21. At If dollars a pound, how many pounds of indigo can be bought for 9f dollars ? 22. How many times is If contained in 9f ? 23. At If dollars a barrel, how many barrels of raisins can be bought for 9f dollars ? 24. How many times is If contained in 9f ? 25. At of a dollar apiece, how many pieces of nankin can be bought for 8| dollars ? 26. How many times is contained in 8f ? 27. At of a dollar a pound, how many pounds of tea can be bought" for 7f dollars ? 28. How many times is f contained in 7 ? 29. How many times is 3^ contained in 7| ? 30. How many times is 5 contained in 17 ? . 31. How many times is 4 contained in 9f ? 32. Plow many times is 3f contained in 12f ? B. 1. At y 1 ^ of a dollar a pound, how many pounds of meat can be bought for of a dollar ? Note. Change ^ to tenths. 2. How many times is yV contained in J ? 8. A man having f of a barrel of flour, distributed it among some poor persons, giving them of a barrel apiece ; hew many did he give it to ? 128 ARITHMETIC. [Par/ 1. Note. CJiange both fractions to twelfths ; that t'j, reduce them to a common denominator. 4. How many times is % contained in J ? 5. If a pound of almonds cost f of a dollar, hovr many pounds can be bought for of a dollar ? Note. Reduce the fractions to a common denomi- nator. 6. How many times is } contained in ? 7. If a piece of nankin cost of a dollar, how many pieces can be bought for 4 dollars ? that is, ^ dol- lars ? 8. How many times is contained in 4f ? 9. If a bushel of barley cost of a dollar, how many bushels can be bought for of a dollar ? How many for 1 f dollars ? 10. How many times is J contained in f ? How many times in If ? 11. How many times is $ contained in ? 12. How many times is contained in f ? TABLES OF COINS, WEIGHTS, AND MEASURES. Ttible I. FEDERAL MONEY. Federal Money is the currency of the IJnited States. 10 mills (m.) make 1 cent, marked ct. 10 cents " 1 dime " d. 10 dimes " 1 dollar, " doll or $ 10 dollars " 1 Eagle, " E. Table II. STERLING MONEY. Sterling Money is the currency of England. 4 farthing? (qr. or far.) make 1 penny, marked d. 12 pence " 1 shilling, . 20 shillings " 1 pound, or a sovereign, " W shillings " 1 guinea. meet. 15.] ARITHMETIC. 129 Table III. TROY WEIGHT. Used in weighing Gold, Silver, and some Liquids. 24 grains (gr.) make 1 pennyweight, marked dwi 20 pennyweights " 1 ounce, " oz. 12 ounces " 1 pound, " Ib Table IV. AVOIRDUPOIS WEIGHT. Used in weighing Groceries, Hay, &c., and all the Metals, except Gold and Silver. It is now the custom to allow 100 Ibs. for a hundred-weight instead of 112 as for- merly. 16 drams (dr.) 16 ounces 25 pounds 4 quarters 20 hundred-weight make 1 ounce, marked oz. 1 pound, " U>. 1 quarter, " qr. " 1 hundred-weight, " cwt. " Iton, " T. Table V. CLOTH MEASURE. Used in measuring Woollens and other Cloths, inches (in.) make 1 nail, 4 nails, or 9 in. 4 quarters 3 quarters 5 quarters 6 quarters marked na. 1 quarter of a yard, " qr. 1 ya*d, " yd. 1 Flemish ell, " Fl e. 1 English ell, " E. e. 1 French ell, " F. e. Table VI. WENE MEASURB. Used in measuring Wines, &c. 4 gills (gl) 2 pints 4 quarts 31 2 gallons 42 gallons 63 gallons, or 2 barrels 2 hogsheads 2 pipes Table Vll. DRY MEASURE. Used in measuring Grains, Salt, Oysters, &c. 2 pints make 1 quart, marked qt. 8 quarts " 1 peck, " pk. 4 pecks " 1 bushel, fo*. ake 1 pint, " 1 quart, " 1 gallon, 1 barrel, 1 tierce, 1 hogshead, 1 pipe or butt, 1 tun, marked pi. qt. " gal bar. or bbl. " tier. " hhd. " pi. " tun. 180 AEITHMETIC. [Part I Table VIII. MEASURE OF TIME. minute, marked min. hour, " hr. day, " dy. k. " wk. wee nion . tn, 60 seconds (sec.) make 60 minutee 24 hours " 7 days 4 weeks " 13 months, 1 day and 6 J hours, or 12 calendar > " 1 year, " yr. months For convenience of reckoning, it is usual in calen- dars to call the year 365 days for 3 successive years, and every fourth year 366, (for in 4 years the six hours overplus amount to a day,) which is called bissextile or leap year. This day is added to Feb- ruary. The common year is divided into twelve months, which are sometimes called calendar months, because they are the months used in calendars. The names of the months, and the number of days in each are as follows : Names. Number of days. 1. January 31 2. February 28, in leap year 29 3. March 31 4. April 30 5. May 31 6. June 30 7. July 31 8. August 31 9. September 30 10. October 31 11. November 30 12. December 31 Miscellaneous Example*. 1. In 2 pounds how many ounces ? 2. In 8 yards how many quarters ? Sect. 15.] ARITHMETIC. 131 3. ID 3 quarters of a yard how many nails ? 4. $j of a dollar is how many cents ? 5. How many farthings is g of a penny ? 6. How many pence is $ of a shilling ? 7. | of a yard is how many quarters and nails ? 8. In f how many shillings ? 9. How much is f of a shilling ? 1 0. How much is f of a bushel of wheat ? 11. How much would f of a barrel of wine cost, at one dollar a gallon ? 12. How much would J cwt. of sugar cost, at 8 cents a pound ? 13. How much is f of a day ? 14. How much is | of a day ? 15. How much is f of a week ? 16. How much is f of an hour ? 17. How much would f of a hogshead of wine cost, at 2 dollars a gallon ? 18. If a man spend 28 dollars in a month, what is that a week ? How much a day ? 19. If a man spend 16 dollars a week, what is that a day? 20. If a man buy 4 bushels of grain for 5 dollars, how much is that a bushel ? 21. If wine is 2 dollars a gallon, how much is that a pint ? 22. If you give 5 cents a gill for wine, what is that a pint ? What is it a quart ? What is it a gal- lon? 23. If wine is worth 20 cents a pint, what is that a gill ? What is it a quart ? What is it a gallon ? 24. If a yard of cloth is worth 7 dollars, what are 2 1 yards worth? 25. If a man earn 11 dollars a week, what is that a day ? What for 3 days ? What for 4 J days ? 26. If a man earn 2f dollars in a day, what will he earn in a week ? 132 ARITHMETIC. [Part 1. 27 What is of a hogshead of wine ? 28. 1 farthing is what part of a penny ? 29. 2 farthings is what part of a penny ? 30. 3 farthings is what part of a penny ? 31. 1 penny is what part of a shilling? 32. 2 pence is what part of a shilling ? 33. 3 pence is what part of a shilling? 34. 5 pence is what part of a shilling ? 35. 6 pence is what part of a shilling? 36. 7 pence is what part of a shilling? 37. 8 pence is what part of a shilling ? 38. 9 pence is what part of a shilling? 39. 10 pence is what part of a shilling ? 40. 11 pence is what part of a shilling? 41. 1 shilling is what part of a pound ? 42. 2 shillings is what part of a pound ? 43. 3 shillings is what part of a pound ? 44. 4 shillings is what part of a pound ? 45. 5 shillings is what part of a pound ? 46. What part of a pound is 6 shillings? 7 shil- lings ? 8 shillings ? 9 shiUings ? 10 shillings ? 1 1 shillings ? 12 shillings ? 13 shillings ? 14 shillings ? I r> shillings ? 16 shillings ? 17 shillings ? 18 shillings ? 19 shiUings? 47. How many farthings are there in a shilling? 48. One farthing is what part of a shilling ? 49. 2 farthings is what part of a shilling ? 3 far- things ? 4 farthings ? 5 farthings ? 6 farthings ? 7 farthings ? 8 farthings ? 9 farthings ? 10 farthings ? 50. How many pence are there in a pound ? 51. One penny is what part of a pound ? 52. What part of a pound is 2 pence ? 3 pence ? 4 pence ? 5 pence ? 6 pence ? 7 pence ? 8 pence : II pence ? 15 pence ? 27 pence ? 35 pence ? 53. How many pence are there in I shilling and 6 pence ? 54. In 2 shillings ard 4 pence, how many pence ? Sect. 15.] ARITHMETIC. 133 55. In 4 shillings and 5 pence, how many ponce ? 56. In 5 shillings and 8 pence, how many pence ? 57. In 9 shillings and 1 1 pence, how many pence i 58. What part of 1 is 2s. 6d. ? 59. 3s. 5d. is what part of 1 ? Note. Reduce the whole to pence. 60. 7s. 8d. is what part of 1 ? 61. What is the price of 2 yards of cloth, at 3s. 4d a yard ? 62. What will 8 yards of cloth cost, at 2s. 8d. a yard ? 63. What will 4 bushels of wheat cost, at 5s. 9d. a bushel ? 64. What must you give for 4 barrels of cider, at 2^ dollars a barrel ? 65. If 3 bushels of wheat be divided between 2 men, how much would they have apiece ? 66. If 4 bushels of corn be divided among 5 men, how much would they have apiece ? 67. If 3 bushels of corn be divided among 7 men, how much would they have apiece ? 68. How many nails are there in 1 yard ? 69. How many nails are there in 4 yards ? 70. How many nails are there in 5 yards and 2 nails ? 71. In 7 yards and 3 quarters, how many quarters ? 72. In 4 yards, 2 quarters, and 3 nails, how many nails ? 73. 1 nail is what part of a quarter ? 74. 3 nails is what part of a quarter ? 75. 1 nail is what part of a yard ? 7 6. What part of 1 yard is 3 nails ? 5 nails ? 7 nails ? 10 nails ? 15 nails ? 77. In 8 quarters of a yard how many yards ? 78. In 12 quarters of a yard how many yards ? 79. In 10 quarters of a yard how many yards ? 80. In 15 quarters of a yard how many yards? 134 ARITHMETIC. [Part 1. 81. In 12 nails how many quarters of a yard ? 82. In 10 nails how many quarters of a yard? How many yards ? 83. In 24 uuils how many quarters of a yard? How many yards ? 84. In 35 quarters of a yard how many yards ? 85. In 45 nails how many yards ? 86. In 63 nails how many yards ? 87. At 2 cents a nail what would 4 yards of cloth cost? 88. At 2 dollars for 1 quarter of a yard, what would 2 yards cost ? 89. 1 oz. is what part of a Ib. ? 90. What part of a Ib. is 2 oz. ? 3 oz. ? 4 oz. ? 5 oz. ? 7 oz. ? 10 oz. ? 15 oz. ? 91. What part of a qr. of 1 cwt. is 1 Ib. ? 2 Ibs. ? 31bs.? 4 Ibs.? 7 Ibs.? 9 Ibs. ? 14 Ibs. ? 18 Ibs.? 23 Ibs. ? 7 29. 30. 81. 32. a3. 84. 500 7667 7543 1812 4648 785 314 1235 1858 1732 4620 2253 409 5555 1776 2712 8665 226 1492 3333 452 45 85. 86. 87. 38. 89. 990 8 6540 38,924 468 6209 295 6030 67,798 26,792 2316 17 2709 54,554 J5,864 2201 7897 5967 33,333 80,112 7. 40. Add together 39, 67, 92, 43, 84, 51. 41. Add together 270, 489, 126, 46. 42. Add together 2368, 235, 496, 1736. 43. Add together 66,470, 801, 4009, 32, 127. 8. 44. A merchant bought 4 pieces of cloth ; for the first piece he gave 53 dollars ; for the second, 78 dollars ; for the third, 68 dollars; and for the fourth, 67 dollars. How much did he give for the whole V 45. In an orchard there are 22 peach-trees, 95 apple- trees, 17 pear-trees, and 56 plum-tivt's. How many trees in the orchard ? 46. A man bought four horses ; for the first he gave 84 dollars; for the second, 150 dollars; for the third, 475 dol- lars; and for the fourth, 526 dollars. How much did he give for all the horses ? 47. In the Boylston school there are 810 scholars; in the Brimmer, 597 scholars; in the D wight, 956 scholars; in the Elliot, 776 scholars; in the Franklin, 572 scholars; and in the Hancock, 731 scholars. How many scholars in these six schools ? 48. A gentleman, building a house, paid the mason 2,964 dollars, the carpenter 5,723 dollars, the plasterer 625 dollars, *nd the painter 354 dollars. How much did the~ whole house cost V [Seepage 173.] ARITHMETIC. 167 SUBTRACTION. 9. 9 less 7 are how many ? This is called a question in Subtraction. We take away or subtract 7 from 9 to find the difference or remainder, which is 2. Ex. 1. From 476 take 245. It is most convenient to write the less number | . j under the greater, units under units, tens under || tens, etc., as in addition. Beginning at the units, 476 we sav ^ from 6 leaves 1, which we write under 245 the units ; then 4 (tens) from 7 (tens) leaves 3 2~Q~~i tens, which we write under the tens ; and 2 (hun- dreds) from 4 (hundreds) leaves 2 (hundreds, which we write down, and the answer is 231. EX. 2. From 939 take 458. * In this example we take 8 from 9 and there j remains 1, which we write down; but since we ||| cannot take 5 (tens) from 3 (tens), we borrow 1 939 (hundred) or 10 (tens) from the 9 (hundreds,) 458 which, joined with the 3 (tens), makes 13 (tens) ; then 5 (tens) from 13 (tons) leaves 8 (tens), which we write down. As one of the 9 (hundreds) has been put with the three (tens), there remains but 8 (hun- dreds). Therefore we say 4 (hundreds) from 8 (hundreds) leaves 4 (hundreds). This, written down, makes the answer 481. 3. 4. 6. 6. 7. 8. 9. From 18 58 35 59 68 82 95 Take 4 6 12 83 34 51 75 Ans. 14 10. 11. 12. 13. 14. 16. From 285 467 9685 7856 8744 2607 Take 74 46 5473 5722 4302 1405 16. 17. 18. 19. 20. 21. 22. From 12 13 19 16 24 56 92 Take 6 5 9 7 8 7 4 168 ARITHMETIC. From 30 Take 20 24. 53 10 26. 46 27 26. 33 25 27. 64 29 92 88 29 84 59 From Take 30. 134 25 31. 244 73 82. 252 171 888 459 34. 2236 1145 35. 3456 2246 86. 9927 6090 37. From 96 take 58. 38. Front 4785 take 8679. 89. From 8344 take 5824. 40. From 10982 take 6470. 41. From 984 take 176. 42. From 4788 take 967. 43. From 7898 take 6898. 44. From 40816 take 5622. 10. 45. In a certain garden there were 86 rose-bushes, but the worms destroyed 83 of them. How many were left ? 46. A man is 66 years of age, and his son 22. How old was the man at the birth of his son V 47. A man gave 240 dollars for his carriage, and 121 dol- lars for his horse. How much more did he give for his carriage than for his horse ? 48. Harry had 84 cents given him to buy some marbles with, but on his way to the store he lost 25 cents. How many cents had he left ? 49. A girl was carrying 72 eggs to market, but she let the basket fall, and 39 eggs were broken. How many had she left? 50. In an army there were 14,642 men, but in a battle 789 men were killed or wounded. How many serviceable men were there left ? MULTIPLICATION. 11. 7 times 5 are how many ? This is called a question in multiplication. To obtain the answer we repeat or wiultipiy the number 5, 7 times ; thus, 7 times 5 are 35. Ex. 1. Multiply 564 by 7. | Writing the numbers thus, the 7 under the | I 3 units, we say, 7 times 4 (units) are 28 (units), or |.g J 2 tens and 8 units. We write down the 8 (units) 564 and reserve the two (tens) ; then, 7 times 6 7 (tens) are 42 (tens) and the 2 (tens) which we 8948 reserved are 44 (tens) or 4 hundreds and 4 tens. We write the 4 (tens) and reserve the 4 (mm- ARITHMETIC. 169 dreds) ; then, 7 times 5 (hundreds) are 35 (hundreds), which, with the 4 (hundreds) we reserved are 39 (hun- dreds) or 3 thousands and 9 hundreds. This we write down, and the answer is 3948. 2. 3. 4. 6. 6. 7. 8. Multiply 8 9 13 23 43 51 65 By 3523697 9. 10. 11. 12. 13. 14. Multiply 143 424 511 821 410 821 By 247654 16. 16. 17. 18. Multiply 813 962 1243 2104 By 3 8 2 4 19. 20. 21. 22. Multiply -34 7110 8947 4355 By 6 7 9 8 23. Multiply 129 by 35. 1 1 Writing the numbers in order, units under units, 1 5" H and tens under tens, we multiply, first, by the 5 1 1 II units, and write the result in its proper place, as j 2 9 when we multiplied by one figure. We multiply o K next by the 3 (tens) and write the first figure of the result in the place of tens, because units mul- 6 4 5 tiplied by tens ought to produce tens. Then add- ?_?_!_ ing together the results obtained by multiplying 4515 by the 5 (units) and 3 (tens), their sum will be 4515, which is the answer. 24. 25. 26 27. 28. 29. 30. Multiply 34 35 48 45 99 612 844 By 23 25 32 44 56 54 29 1Q2 68 Ans. 782 170 ARITHMETIC. 31. 32. 83. 34. 85. 86. 87 Multiply 676 434 467 548 1835 2972 5963 By " 76 322 212 403 768 256 4678 38. Multiply 816 by 6. 89. Multiply 363 by 48. 40. Multiply 846 by 65. 41. Multiply 152 by 87. 42. Multiply 2523 by 47. 43. Multiply 682 by 746. 44. Multiply 8422 by 186. 45. Multiply 3107 by 761. 46. How much will 48 barrels of flour cost at 8 dollars a barrel ? 47. In a room there are 9 windows with 16 panes of glass in each. How many panes of glass in all the win- dows ? 48. If a man receive 47 dollars a month, how much will he receive in twelve months ? 49. How much will 126 tons of hay cost, at 13 dollars a ton? 50. There are 365 days in a year. How many days in 19 years? [See page 175.] DIVISION. 12. 24 are how many times 8 ? This is called a ques- tion in division, because we divide or separate the larger number into as many equal parts as there are units in the smaller number. 13. The number to be divided is called the dividend , the number by which we divide, the divisor ; and the result, the quotient. If anything is left after the division, it is called the remainder. 14. Ex. 1. Divide 425 by 5. 4 in ,,Uo,,5)425Dmdend. Wrfte d WD the div!(len<1 nd 8 5 Quotient, divisor 5 at the left of the dividend, and draw a curved line between them. First find how many times 5 is contained in 42 (tens). It is contained 8 ARITHMETIC. 171 (tens) times with 2 (tens) remaining. We write down the 8 (tens) undei the tens and suppose the remainder 2 (tens) written at the left of the next figure of the dividend mak- ing the number 25. 5 is contained in 25, 5 times without remainder. We write the 5 underneath and the division is completed. Division performed in this way is called short division. 15i Perform in this way the following examples : 2. 3. 4. 5. 6. 7. Div*K.3)9 Divid.nd. ?)26 8)36 4)88 5)15 8)18 8 (luotient. 8. 9. 10. 11. 12. 13. 7)21 9)36 6)48 8)72 9)81 4)40 14. 15. 16. 17. 18. 6)7-2 2)444 4)256 3)963 7)847 19. 9)279 20. 5)6825 21. 6)4650 22, 9)^72 23. 4)4484 24. 8)1312 16. When the divisor consists of two or more figures, it is more convenient to perform the operation in the following manner : 25. Divide 354 by 13. , 1 i .5 S .5 Jg piYisor. 13)3 5 4 (2 7 We write the dividend and draw a 2 6 curved line on both sides of it. We 9 4 first inquire how many times 13 is 9 i contained in 35 (tens) the first two 3 r figures of the dividend. We find that it is contained 2 (tens) times. Place the 2 (tens) at the right of the dividend as the first figure of the quotient, and then multiplying the divisor 13 by the 2 tens, write the product 26 (tens) under the 35 (tens) of the dividend, and subtract. To the remainder 9 172 ARITHMETIC. (tens) we annex 4, the next figure of the dividend, anC find how many times the divisor 1 3 is contained in 94. It is contained 7 times. Write the 7 (units) at the right of the 2 (tens) in the quotient and multiply the divisor by the 7 (units). Write the product 91 under the 94 and sub- tract, and there is left a remainder of 3. 13 is therefore contained in 354 27 times with 3 remainder. Division performed in this way is called long division. IT. Perform in this way the following examples : 26. 27. 28. 29. 13)286(22 15)235( 17)425( 13)585( 26 "26 26 8o7 81. 82. 83. 16)1168( 13)676( 12)1728( 15)2445( 84. 35. 36. 37 3)301G( 17)2057( 19)2994( 15)4445( 38. Divide 144 by 12. 89. Divide 225 by 15. 40. Divide 180 by 12. 41. Divide 5456 by 22. 42. Divide 1411 by 36. 43. Divide 8844 by 22. 44. How much sugar at 9 cents a pound can be bought for 81 cents? 45. If an acre of land cost 13 dollars, how many acres can you buy for 351 dollars ? 46. If a man can walk 21 miles in a day, how long will it take him to walk 273 miles ? 47. A man divided 23,796 dollars among his 6 children. How much did each receive ? 48. A railroad company employed 65 workmen, and at the end of the month it took 1755 dollars to pay them. How much did each man receive ? 49. There are 24 hours in one day ; how many days are there in 1656 hours ? 50. At 97 dollars a ton, how many tons of iron may be bought for 8439 dollars. [See page 175.] ARITHMETIC. 173 TABLES OF PROBLEMS FOR THE BLACKBOARD, ADDITION. These Tables are designed to furnish a number of problems for solu- tion on the Blackboard. They can be used in a great variety of ways ; thus in the table for Addition the numbers can be added as they stand ; two or more can form a problem ; the short columns can be added, &c. 1. 564 + 731 + 401 + 771 + 118 2. 230 1210 261 3542 1001 3. 1348 8625 4010 300 2071 4. 4689 14807 309 5795 2100 6. 394 68175 92 7631 7920 6. 6752 8762 5861 387 1927 7. 397 93721 3872 5961 6092 8. 9687 976 6872 4617 7764 9. 2346 56821 583 3846 389 10. 3741 468 468 586 9999 11. 287 79276 1684 592 1000 12. 1692 68340 587 48009 6130 13. 768 6782 4632 3874 5959 14. 483 483 5879 6879 1527 15. 5068 38192 671 287 8065 16. 473 628 596 7961 6099 17 786 9172 4869 326 684 18. 307 791 8734 4891 1888 19. 5961 998 96871 6873 3039 20. 786 7862 6782 592 5757 21. 597 5917 4839 8783 6495 22. 8643 1010 609 6792 7241 23. 7995 6783 4687 958 5894 24. 5269 9605 8567 2799 8050 25. 2709 32699 9248 5484 909 26. 9628 6183 5304 7640 9876 27. 6825 8744 9989 2007 74361 28. 9927 4302 5673 8376 67945 29. 6728 5722 3565 9090 46839 80. 2786 7856 4077 7365 59876 174 ARITHMETIC. TABLES OF PROBLEMS FOR THE BLACKBOARD. SUBTRACTION. FIND THE DIFFERENCE BETWEEN I, 2. - Si and u 452 695 29. 30. 38765 and 46872 and u 48687 36872 3. 4. 932 684 H H 1453 1596 31. 32. u 2983 68741 u tt 876 19234 5. 6. 888 1010 M 937 1629 33. 34. 59687 38794 11 u 3872 68704 7. 8. 554 923 H 632 451 35. 36. 65823 168721 n 38726 59233 9. 10. M 909 375 U M 1118 4286 37- 38. 59687 38726 u u 26870 9862 11. 12. , 2203 4224 U U 997 325 39. 40. 40687 56043 u 1134J 687* 13. 14. u 409 5376 u H 6102 7635 41. 42. 261807 592687 u M 3872* 15. 16. u 6001 3556 u M 601 4937 43. 44. 38740 93341 u u 587? 68743 17. 18. i 1204 5872 U U 9005 3478 45. 46. 67215 164327 u u 98672 19872 19. 20. , 6226 3248 M M 4195 8423 47. 48. a 6824 93678 u M 6875 3268 21. 22. u 8049 6412 U u 4901 1010 49. 50. 10000 10982 u u 999S 64 7k 23. 24 ; 1965 3824 u u 3642 4007 51. 52- 79797 40815 u 68888 56225 25. j 9999 8763 u u 1098 2854 53. 54. 66476 56324 u u 58134 83459 27. 28. t 5542 1537 u 14 998 3336 55. 56. 789859 984967 a U 47885 736684 ARITHMETIC. 175 TABLES OF PROBLEMS FOR THE BLACKBOARD. MULTIPLICATION. These numbers are to be multiplied by 26, 37, 48, 69, 327, 468, 589, 376 Uso the numbers in one column by those in the other columns. 1. .37 17. 2900 33. 4101 49. 6954 65. 62437 2. 42 18. 3305 34. 3762 50. 4687 66. 88106 3. 56 19. 9999 35. 6256 51. 6492 67. 27091 4. 81 20. 5678 36. 7815 52. 9070 68. 78305 5. 253 21. 3255 37. 5942 53. 46872 69. 64092 6. 163 22. 2247 38. 8307 54. 35906 70. 38764 7. 813 23. 8770 39. 9525 55. 43298 71. 10001 8. 340 24. 4468 40. 5015 56. 39612 72. 98271 9. 2518 25. 3173 41. 2405 57. 47385 73. 64327 10. 1130 26. 6060 42. 1000 58. 96826 74. 86475 11. 4008 27. 9318 43. 1710 59. 13478 75. 90090 12. 3735 28. 2165 44. 6341 60. 26593 76. 145301 13. 4265 29. 2924 45. 1234 61. 80435 77. 309075 14. 1704 30. 3273 46. 8888 62. 72461 78. 900000 15. 8700 31. 1702 47. 1005 63. 38675 79. 946871 16. 7100 32. 5218 48. 8095 64. 92413 80. 362432 DIVISION. To be divided by such numbers as the Teacher may designate 1. 462 17. 847 33. 26830:49. 90684 65. 826541 2. 924 18. 8866 34. 86109150. 43592 66. 487455 3. 796 19. 4595 35. 92165 51. 30030 67. 305095 4. 566 20. 3796 36. 42061 52. 50505 68. 874687 5. 248 21. 7643 37. 56824 53. 906870 69. 604090 6. 806 22. 9262 38. 41837 54. 765903 70. 268405 7. 749 23. 2673 39. 74587J55. 987648 71. 368456 8. 8181 24. 7924 40. 90411 ; 56. 264371 72. 461043 9. 4848 25. 5050 41. 74261| 57. 874687 73. 782654 10. 9635 26. 7735 42. 83501 58. 682409 74. 387005 11. 3721 27. 26874 43. 96843 59. 360002 75. 100101 12. 9900 28. 54672 44. 16890 60. 940856 76. 961147 13. 4805 29. 87964 45. 26874 61. 604379 77. 570880 14. 4650 30. 70865 46. 38421 62. 200648 78. 306890 15. 9972 31. 86304 47. 92674 63. 961478 79. 560437 16. 6825 32. 72591 48. 58435 64. 248700 80. 818127 176 AMTHMETIC. In the study of Mental Arithmetic, too much stress can hardly be put upon the importance of an intimate familiarity with the tables of Ad- dition, Subtraction, Multiplication, &c. On this account it has been thought best to insert here the Multiplication Table, which may be re- ferred to with profit in the earlier lessons of the book, and until, in all ite combinations, it is entirely familiar to the young learner. MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12 2 ' 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 To find how many 8 times 6 are, put your finger on the 8 in the left-hand column ; then move the finger to the right till it comes under the 6 in the top line, and there you find 48, which is 8 times 6. In the same manner learn the whole table. PICCIOLA. By X. B. Saintine. A new translation from the Author's last revised edition. With Illustrations on wood, from designs by Leo- pold Flameng, and ornamental head and tail pieces. Fine edition. In one volume, crown 8vo. Price in extra cloth, gilt, $1.75; half Rox- burgh, $4.50 ; Turkey morocco, $6.00. 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