Harvard college LIBRARY THEE SS E X( J JnJ SXITUT E TEXT-BOOK COLLECTION GIFT OF GEORGE ARTHUR PLIMPTON OF NEW YORK JANUARY 25, 1924 IN ARITHMETIC, oxr THX PLAN OF PESTALOZZL WITH SOME IMPROVEMENTS. BY WARREN COLBURN. STEREOTYPED BY T. H. CARTER & CO. BOSTON. Boston: PUBLISHED BY CUMMINGS, HILLIARD, & Ca Sold BooksellerS generally throughout tJie country, 1825. Sir, DISTRICT OF MASSACHUSETTS, TO WIT: District Clerk's office. BE it remembered, that on the fifth day of April, A. D. 1822, in the Forty-sixth year of the Independence of the United States of America, Cummings Hilliard, of the said District, have deposited in this of- fice the title of a Book, the right whereof they claim as Proprietors, in the words following, to wit: "First Lessons in Arithmetic on the plan of Pestalozzi. With some improvements. By Warren Colburn. Stereotype Edition." 'In conformity to the Act of the Congress of the United States, en- titled, " An Act for the encouragement of Learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such •.copies during the times therein mentioned and also to an Act entitled, "An Act supplementary to an Act, entitled, An Act for the encourage- ment of learning, by securing the copies of maps, charts, and books, to the authors ana proprietors of such copies during the times therein mentioned; and extending the benefits thereof to the arts of Designing, Engraving, and Etching Historical, and other prints." JOHN W. DAVIS, Clerk of the District of Massachusetts. RECOMMENDATIONS. Sir, Boston, 15 November, 1821. I have made use of the Arithmetic and Tables, which you sometime since prepared, on the system of Pestalozzi; and have been much gratified, with the improved edition of it, which you have shown me. I am satisfied from experiment, that it is the most effectual and interesting mode of teaching the science of numbers with which I am acquainted. Respectfully, your obedient servant, HENRY COLMAN. *Mr. Warren Colburn. Having been made acquainted with Mr. Colburn's treatise on Arithmetic, and having attended an examination of his scholars who had been taught according to this system, I am well satisfied that it is the most easy, simple, and natural way of introducing young persons to the first principles in the science of numbers. The method here proposed is the fruit of much study and reflection. The author has had considera- ble experience as a teacher, added to a strong interest in the subject, and a thorough knowledge not only of this but of many of the higher branches of mathematics. This little work is therefore earnestly recommended to the notice of those who are employed in this branch of early instruction, with the belief that it only requires a fair trial in order to be fully ap- provedjmd adopted. J. FARRAR, /^5^V Prof. Math. Harvard University. jfimd&ge, Nov. 16,1821. ^ harvardcfiLMimm JANUARY 25, 1024 PREFACE. As soon as a child begins to use his senses, nature contin- ually 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 learn- ed 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; V hence they will often perform these operations without any previous instruction. If, for example, one child has three ap- £ pies, and another five, they will readily tell how many they both have; and how many one has more than the other. If a :jj child be requested to bring three apples for each person in the room, he will calculate very readily how many to bring, if the number does not exceed those he has learnt. Agaiu, if a child be requested to divide a number of apples among a certain number of persons, he will contrive a way to do it, and will tell how many each must have. The method which children take to do these things, though always correct, is not always the most expeditious. The fondness which children usually manifest for these exercises, and the facility with which they perform them; seem to indicate that the science, of numbers, to a certain ex- tent, should be among the first lessons taught to them * ^ To succeed in this, however, it is necessary rather to fur- nish occasions for them to exercise their own skill in per- forming examples, than jto give them rules. They should be allowed to pursue their own method first, and then they should be made to observe and explain it, and if it was not * See on this subject two essays, entitled Juvenile. Studies, in the Prize Book of the Latin school, Nos. I and II. Published by Cum- mings & Hilhard, 1820 and 1821. tv PREFACE. the best, some improvement should be suggested. By fol- lowing this mode, and making the examples gradually in- crease in difficulty; experience proves, that, at an early age, children may be taught a great variety of the most useful combinations of numbers. Few exercises strengthen and mature the mind so much as arithmetical calculations, if the examples are made sufficient- ly simple to be understood by the pupil; because a regular, though simple process of reasoning is requisite to perform them, and the results are attended with certainty. The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to all things with which we are acquainted, we obtain an abstract idea of number. We first make calculations about sensible objects; and we soon observe, that the same calculations will apply to things very dissimilar; and finally, that they may be made without reference to any particular tilings. Hence from particulars, we establish general principles, which serve as the basis of our reasonings, and enable us to proceed step by step, from the most simple to the more com- plex operations. It appears, therefore,, that mathematical reasoning proceeds as much upon the principle of analytic induction, as that of any other science. Examples of any kind upon abstract numbers, are of very little use, until the learner has discovered the principle from practical examples. They are more difficult in themselves, for the learner does not see their use; and therefore does not so readily understand the question. But questions of a practical kind, if judiciously chosen, show at once what the combination is, and what is to be effected by it. Hence the pupil will much more readily discover the means by which the result is to be obtained. The mind is also greatly assist ed in the operations by reference to sensible objects. When the pupil learns a new combination by means of abstract ex- amples, it very seldom happens that he understands practical examples more easily for it, because he does not discover the connexion, until he has performed several practical exam- ples and begins to generalize them. After the pupil comprehends an operation, abstract exam- , pies are useful, to exercise him, and make him familiar with it. And they serve better .to fix the principle, because they teach the learner to generalize. From the above observations, and from his own experi- ence, the author has been induced to publish this treatise; in which he has pursued the following plan, which seemed to him the most agreeable to the natural progress of the min,d. PREFACE. GENERAL VIEW OF THE PLAN. Every combination commences with practical examples. Care has been taken to select such as will aptly illustrate the combination, and assist the imagination of the pupil in per- forming it. In most instances, immediately after the prac- tical, abstract examples are placed, containing the same numbers and the same operations, that the pupil may the more easily observe the connexion. The instructer should be careful to make the pupil observe the connexion. After these are a few abstract • examples, and then practical ques- tions again. The numbers are small, and the questions so simple, that almost any child of five or six years old is capable of under- standing more than half the book, and those of seven or eight years old can understand the whole of it. The examples are to be performed in the mind, or by means of sensible objects, such as beans, nuts, &c. or by means of the plate at the end of the book. The pupil should first perform the examples in his own way, and men be made to observe and tell how he did them, and why he did them so * * It is remarkable, that a child, although he is able to perform a va- riety of examples which involve addition, subtraction, multiplication, and division, recognises no operation but addition. Indeed, if we ana- lyze these operations when we perform them in our minds, we shall nnd that they all reduce themselves to addition. They are only differ- ent ways of applying the same principle. And it is only when we use an artificial metnod of performing them, that they take a different form. If the following questions were proposed to a child, his answers would be, in substance, like those annexed to the questions. How much is five less than eight? Ans. Three. Why? because live and three are eight. What is the difference between five and eight? Ans. Three. Why? because five and three are eight. If you divide eight into two parts, such that one of the parts may be five, what will the other be? Ans. Three. Why? because five and three are eight. How much must you give for four apples at two cents apiece? Ans. Eight cents. Why? because two and two are four, and two are six, and two are eight. How many apples, at two cents apiece, can you buy for eight cents? Ans. Four. Why? because two and two are four, and two are six, and two are eight. We shall be further convinced of this if we observe that the same table serves for addition and subtraction; and another table which is 1* vi PREFACE. The use of the plates is explained in the Key at the end of the book. Several examples in each section are performed in the Key, to show the method of solving them. No answers are given in the book, except where it is necessary to explain something to the pupil. Most of the explanations are given in the Key; because pupils generally will not understand any explanation given in a book, especially at so early an age. T.he instructer must, therefore, give the explanation viva voce. These, however, will occupy the instructer but a very short time. The first section contains addition and subtraction, the sec- ond multiplication. The third section contains division. In this section the pupil learns the first principles of fractions and the terras which are applied to them. This is done by making him observe that one is the half of two, the third of three the fourth of four, &c. and that two is two thirds of three, two fourths of four, two fifths of five, &c. The fourth section commences with multiplication. In this the pupil is taught to repeat a number a certain number of times, and a part of another time. In the second part of this section the pupil is taught to change a certain number of twos into threes, threes into fours, &c. In the fifth section the pupil is taught to find J> i> &c- and §, |, j, &c. of numbers, which are exactly divisible into these parts. This is only an extension of the principle of frac- tions, which is contained in the third section. In the sixth section the pupil learns to tell of what number any number, as 2, 3,4, &c. is one half, one third, one fourth, &c.; and also, knowing f, f, f, &c. of a number, to find that number. These combinations contain all the most common and most useful operations of vulgar fractions. But being applied 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 arc so arranged, that al- most any child of six or seven years old will readily compre- hend them. And the questions are asked in such a manner, that, if the instructer pursues the method explained in the Key, it will be almost impossible for the pupil to perform any example without understanding the reason of it. Indeed, in formed by addition, serves both for multiplication and division. In this treatise the same plate serves for the four operations. This remark shows the necessity of making the pupil attend to his manner of performing the examples and of explaining to him the dif- ference between them. PREFACE. 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 ot the book. These principles are sufficient to enable the pupil to per- form almost all kinds of examples that ever occur. He will not, however, be able to solve questions in which it is neces- sary to take fractional parts of unity, though the principles are the same. After section sixth, there is a collection of miscellaneous examples, in which are contained almost all the kinds that usually occur. There are none, however, which the prin- ciples explained are not sufficient to solve. In section eight 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 operations do not differ materially from those in the preceding sections. There are some operations, however, peculiar to fractions. The two last plates are used to illustrate 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 operations may be performed in the mind. Afterwards he has only to learn the application of figures t<5 these operations, and his knowledge of arithmetic will be complete. The Rule of Three, and all the other rules which are usu- ally contained in our arithmetics, will be found useless. The examples under these rules will be performed upon general principles with much greater facility, and with a greater de- gree of certainty. The following are some of the principal difficulties which a child has to encounter in learning arithmetic, in the usual way, and which are seidom overcome. First, the exam- ples 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. This increases the difficulty very much, for even if the numbers were so small, that the pupil could comprehend them, he would dis- cover but very little connexion between them, and practical examples. Abstract numbers, and the operations upon them, must be learned from practical examples; 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 ex- pressed by figures, which, if they were used only as a con- viii PREFACE. tiacted way of writing numbers, would be much more dif- ficult to be understood at first, than the numbers written at length in words. But they are not 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 commences arithmetic is present- ed with a set of abstract numbers, written with figures, 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 signify, and what is meant by addition, subtraction, multiplication, 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 perform 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 general principle; but with respect to him, and often times to the instructer himself it is so far from it, that it hardly deserves to be called even a mechanical principle. He performs the ex- amples, and makes the answers agree with those in the book, and so presumes they are right. He is 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 mistrust they belonged to it. But finding them there, he applies his rule to them, and obtains the an- swers, 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 opera- tions necessary to solve it; but in comparing it with the exam pies which he has performed before, and endeavouring to dis- 7 PREFACE. ix cover some analogy between it and them, either in the sound, or in something else. If he is fortunate 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 ex- perienced more or less inconvenience from this mode of pro- ceeding. Almost all, who have ever fully understood arith- metic, have been obliged to learn it over again in their own way. And it is not too bold an assertion to say, that no man ever actually learned mathematics in any other method, than by analytic induction; that is, by learning the principles by tne examples he performs; and not by learning principles first, and then discovering by them how the examples are to be performed. In forming and arranging the several combinations the au- thor has received considerable assistance from the system of Pestalozzi. He has not however had an opportunity of seeing Pestalozzi's own work on this subject, but only a brief outline of it by another. The plates also are from Pestalozzi. In selecting and arranging the examples to illustrate these com- binations, and in the manner of solving questions generally, he has received no assistance from Pestalozzi. THE BOY WITHOUT A GENIUS. Mr. ^iseman, the schoolmaster, at the end of his sum- mer 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 elev- en, and yet can do nothing but read bis mother tongue, aud that but indifferently. We sent him at seven to a grammar school in our neighbourhood; but his master soon found that his genius was not turned to learning languages. He was then put to writing, 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 either. He could do nothing ia geography for want of memory. In short, if X PREFACE. 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 favoured shortly with your opinion about him, and remain, sir, our most obedient servant, HUMPHREY ACRES. When Mr. Wiseman had read this letter he shook his head, and said T 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 any thing about it—no uncommon error! Let us see, however, what the vouth looks like. I suppose he is a human creature at least. Master Samuel Acres was now called in. He came hanging down his head, and looking as if he was going to be flogged. 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? Yes, sir. Can you write? Samuel 7 I learned it a little, sir, but I left it off again. And why so? Because 1 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. Samuel held out both his'paws, like a dancing bear. 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 JIAS 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 any thing 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. PREFACE. U Yes, sir, I know them. And the months in the year, perhaps. Yes, sir. And you could probably repeat the names of your brothers and sis- ters, and all your father's servants, and half the people in the village besides. I believe I could, sir. Well—and is hie, iuec, hoc, more difficult to remember than these? Samuel was silent. * Have you learned any thing 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 1 Six. And how many for two-pence 1 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 awhile, and then said, five-pence. Right Why here you have been practising the four great rules of arithmetic, addition, subtraction, multiplication, and division. Learn- ing 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 J can't nere. 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 call- ed out to set him a copy of letters, and another was appointed to hear him in grammar. He read a few sentences in 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 ta- ble? when Mr. Wiseman thought it time to answer his father's letter; which he did as follows: Sir, I now think it right to give you some information concerning your son. You perhaps expected it sooner, btV. I always wish to avoid nasty 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 to any one pursuit as will lead to excel with little or no labour or instruction, I must say that I have not met with 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 man- PREFACE. kind can do when properly taught, I can affirm, that I find in him no peculiar deficiency. And whether you choose to bring him up to trade or to some practical profession, I see no reason to doubt that he may in time become sufficiently aualified for it. It is my favourite maxim, sir, that every thing most valuable in this life may generally be acquir- ed 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 any thing of their own accord but a top or a marble. IVill take care while he is with me that he loses no more time this way, but is employed about things 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 with Mr. Acres' notions, yet being convinced that Mr. Wiseman was more like- ly 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. t In due time a profession was chosen for him, which seemed to suit his temper and talents, but for which he 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. I 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 1 5. If you count the thumb with the fingers, how many will it make 1 6. If you shut your thumb and one finger and leave the rest open, how many will be open 1 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 7 9. If you count all the fingers on one hand, and two on the other, how many will there be 1 10. George has three cents, and Joseph has four; how many have they both together? * For the manner of solving questions, and the explanation of the plates, see the key at the end of the book. The first questions in this section are intended for very young children. It will be well for the instructer U> give a great m&of more of this Jrind.-—Older pupils may-omit these. 2 14 ARITHMETIC. [Part U 11. Robert gave five cents for an orange, and two for an apple, how mailj did he give for both 1 12. If a custard cost, six cents, and an apple two cents; how many cents* will it take to buy an appje and a custard 1 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 1 how 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? 16. How many fingers have you on both hands 1 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 1 how many would you have in both hands 1 19. David had seven nuts, and gave three of them to George, how many had he left? 20. Two boys, James and Robert, played at mar- bles; when they began, they had seven apiece, and when they had done, James had won four; 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 you 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 1 24. A man bought a barrel of flour for eight dol- lars, and sold it for four dollars more than he gave for it; how much did he sell it for 1 Sect. L] ARITHMETIC. 15 25. A man bought a hundred weight of sugar for nine dollars, and a barrel of flour for seven dol- lars, how much did he give for the whole 1 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 ™ dollars, but, being damaged, he sold it again for eight dollars; how much did he lose 1 28. A man bought three sheep for fifteen dollars, but could not sell them again for so much by eight 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 1 . 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 1 31. A man owing fifteen dollars, paid nine dol- lars of it, how much did he then owe 1 32. A man owing seventeen dollars, paid all but seven dollars; how much did he pay? B. 1. Two and one are how many 1 2. Two and two are how many? 3. Three and two are how many 1 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 aire how many I 8. Eight and two are how many t 9. Nine and two are how many % 10. Ten and two are how many 1 11. Two and three are how many? 12. Three and three are how many t 13. Four and three are how many 1 14. Five and three are how many 1 16 ARITHMETIC. 15. Six and three are how many? § 16. Seven and three are how many 17. Eight and three are how many 18. Nine and three are how many 1 19. Ten and three are how many 1 20. Two and four are how many? 21. Three and four are how many? 22. Four and four are how many? 23. Five and four are how many 1 24. Six and four are how many 1 25. Seven and four are how many 1 26. Eight and four are how many 1 27. Nine and four are how many 1 28. Ten and four are how many 1 29. Two and five are how many? 30. Three and five are how many? 31. IJour and five are how many? 32. Five and five are how many 1 33. Six and five are how many? 34. Seven and five are how many 7 35. Eight and five are how many 1 36. Nine and five are how many? 37. Ten and five are how many 1 38. Two and six are how many? 39. Three and six are how many? 40. Four and six are how many 1 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 1 46. Ten and six are howmany 1 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 1 51. Six and seven are how many? Sect. 1.] ARITHMETIC. 21 10. A man bought a sleigh for seventeen dollars, and gave nine dollars to have it repaired and paint- ed, and then sold it for twenty-three dollars; how much did he lose by the bargain? 11. Eleven and two are how many? 12. Eleven and three are how many? 13. Eleven and four are how many? 14. Eleven and five are how many? 15. Eleven and six are how many? 16. Eleven and seven are how many? N^T- Eleven and eight are how many? lfether, four; how far apart will they be in one houf? How far in two hours? How far in four hours 11 3®. If three, men can wo a piece of work in two A ARITHMETIC. [Flirt 1. 0 u days, how many days would it take one man to do it? 40. If four men can do a piece of work in five days, how many days would it take one man to do it? 41. If six men can do a piece of work in seven days, how many men would it take to do it in one day? 42. If a quantity of provisions will serve three men five days, how many men would it serve one day? 43. If a quantity of provisions will serve five men seven days, how many days would it serve one man? 44. If fifteen dollars worth of provision will serve eight men five days, how many days will it serve one man? 45. A man had a piece of work to perform which seven men could do in nine days, but it was neces- sary that the whole should be completed in one day; how many men must he employ? 46. If the interest of one dollar is six cents a year, what would be the interest of ten dollars for the same time? 47. If the interest of one dollar is six cents for one year, what would be the interest of it for two years ? for three years? for seven years? 48. If a man can earn seven shillings in a day, how many shillings will he earn in six days? 49. If a man can earn eight dollars in a month, how much can he earn in si^: months? 50. At five dollars a week:, what will nine weeks board come to? I £) 0' 51. A lady bought threes yards of cambric at two dollars a yard, seven yards/of silk for three dollars a yard, five yards of riband for four dollars, and some crape for two dollars; she Amid four ten-dollar bills how much* must she recei/e back again? Page Missing in Original Volume Page Missing in Original Volume Page Missing in Original Volume Page Missing in Original Volume 8ect.&] ARITHMETIC. 46 lings, how much can you buy for five shillings? How much fornix shillings 1 How much for seven ^ shillings 7 * 43. Five are how tnany times four? Arts. Once four and one fourth of four 44. Six are how many times four 1 Arts. Once four, and two fourths of four. 45. Seven are how many limes four? Ans. Once four, and three fourths of four. 46. Eight are how many times four 1 2, 47. If four bushels of corn will buy one yard of cloth, how many yards will nine bushels buy 1 How^£ many yards will ten bushels buy^ How maay ^ards will eleven bushels buy ?^Ji tf * ) 48. What do you understand by one fourth, two-; fourths, or three fourths of any thing? See remark after example 16th. 49. Ten are how many times four 50. Eleven are how many times four &!jfL 51. Twelve are how many times four T% [ < 52. Thirteen are how many times fourf)^ ^ 53. Fourteen are how many times four!«ML£^ 1 54. Fifteen are how many times four 7*5 ^ 55. Sixteen are how many times four ?'^T 56. If a barrel of flour be worth five dollars, a&d it be divided equally among five men, what will onefe man's share be worth 1 that is, what is one fifth of barrel worth V What are two fifths of it worth Wc+ What are three fifths of it worth t?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 bu^^What part*- will two dollars buy^&What part will three dollars ' j buy fitWhat part will four dollars buy *j \ 58/ What part of five is one? ¥ W Ans. One is the fifth part of five. 59. Two is what part of five 1 1| 4 46 ARITHMETIC. [Part I. Arts. Two fifths of five. "60. Three is what part of five? Arts. Three fifths of five. 61. Four is what part of five 1 62. How many fifths make a whole one 1 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 1 How many for thirteen cents? 64. What do you understand by one fifth, two fifths, &c. of any thing 1 See remark after example 16th. 65. Seven are how times five? Am. Once five and two fifths of five. 66. Eight are how many times five 1 67. Nine are how many times five? 68. Ten are how many times five? 69. Eleven are how many times five 1 70. Twelve are how many times five 1 71. Thirteen are how many times five? .72. Fourteen are how many times five 1 73. Fifteen are how many times five 1 74- M a barrel of beef cost six dollars, and it were dpided 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 1 Four sixths 1 Five sixths? 75. If fish is worth six dollars a barrel, what part of a barrel will one dollar buy? What part of a bar- rel will two^ dollars buy? Three dollars 1 Four dol- *iff§3 Five dollars? ($6. What part of six is one*? Ans. On 40. Fifty are how many times 9? 5? 4 1 10f? 8? 617? • Sect. 3.] ARITHMETIC. $7 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 1 .31. There is a vessel containing sixty-three gal- lons of wine; it has a pipe which discharges 7 gal- lons in an hour; how many hours will it take to empty the vessel 1 32. There is a vessel containing eighty-seven gallons, and by a cock ten gallons will run into it in an hour; in how many hours will the vessel be filled 1 33. If one man can do a piece of work in thirty <^vs, in how many days can 3 men doit 1 in how iSflny days can 5 men do it 1 34. If you wish to put sixty-four pounds of but- ter into 8 boxes, how many pounds would you put into each box 1 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 1 36. If a man can perform a journey in thirty*six hours, how many days will it take him to do it when the days are nine hours long 1 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 1 63 ARITHMETIC. [Part 1. 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 3 dollars a yard, what will 4 yards and 1 third of a yard of cloth cost? 4. 4 times 3, and 1 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 1 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 cnst 1 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 5, 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? ^ 16. 4 times 6, and 1 sixth of 6 are how many? 17. 7 times, 6, and 5 sixths of 6 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 b.arrel cost? ^~ 19. 3 times 7, and 1 seventh of 7 are how many? t \ 20. 5 times 7, and 2 sevenths of 7 are how many 1, Sect. 4.] ARITHMETIC. 59 21. 8 times 5, and 4 fifths of 5 are how many? 22. 8 times 6, and 3 sixths of 6 are how many 1 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 1 26. 8 times 7, and 4 sevenths of 7 are how many? 27. 9 times 7, and 6 sevenths of 7 are how many 1 28. 3 times 8, and 5 eighths of 8 are how many? 29. 9 times 8, and 7 eighths of 8 are how many 1 30. If a hundred weight of sugar cost 9 dollars, what will 2 hundred weight and 1 ninth of a hun- dred 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 1 32. 5 times 9, and 2 ninths of 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 1 35. 7 times 9, and 7 ninths of 9 are how many 1 36. 5 times 10, and 4 tenths of 10 are how many 1 37. 8 times 9, and 5 ninths of 9 are how many 1 38. 4 times 10, and 7 tenths of 10 are how many? 39. 6 times 10, and 9 tenths of 10 are how many 1 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 1 ->w 2. 2 times 6 are how many times 41 3. A man bought 3 yards of cloth at 4 dollars a yard, how many dollars did it come to 1 fltow much flour at 6 dollars a barrel would it take to pay for it 1 4. Z times 4 are how many times 6 1 5. A man bought 4 peaches at 3 cents apiece, how many cents did they come to? he paid for 60 ARITHMETIC. [Part 1. them with pears at 2 cents apiece, how many pears did it take 1 ti, 4 times 3 are how many times 2? 7. Bought 2 hundred weight of sugar, at 9 dol- lars a hundred weight, and paid for it with wood at 6 dollars a cord; how many'cords did it take 1 8. 2 times 9 are how many times 61 9. Bought 3 barrels of flour at 8 dollars a barrel, and paid for it with cider at 4 dollars a barrel; how many barrels did it take 1 10. 3 times 8 are how many 'dmes 4 7 11. 12 times 3 are how many times 5 7 12. 6 times 4 are how many times 8 1 13. 3 times 10 are how many times 6 1 14. 4 times 9 are how many times 6 1 15. How much flannel worth 4 shillings a yard, must be given for 3 yards of silk worth 5 shillings a yard 1 16. 3 times 5 are how many times 4 1 17. 2 times 7 are how many times 3? 51 4 7 18. 4 times 5 are how many times 3 7 6 7 71 19. 3 times 7 are how many times 4 7 57 6 7 87 9? 20. Bougnt 2 kegs and 2 sevenths of a keg of tobacco at 7 dollars a keg, and paid for it with wood at 4 dollars a cord; how many cords did it take 7 How much butter at 3 dollars a box would it take to pay for it 7 21. 2 times 7, and 2 sevenths of 7 are how many times 4? 3 7 5 1 61 8? 32. 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 1 23. 3 times 5, and 3 fifths of five are how manv times 6? 9? 4? 71 31 81 24. How much sugar that is 8 dollars a hundred i half of 8? you divide 8 apples equally among 4 boys, hat part of them must each have? Ans* One fourth of them. 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 1 £yg£hat is 1 third of 6? pWJ^ yards of cloth cost 8 dollars, what part of 8 dollars would 1 yard cost? What part of 8 dollar®^ J would 2 yards cost? What part of 8 dollars would yards cost? ip 10. What is 1 fourth of 8? What is 2 fourths 81 What is 3 fourths of 8? . <|& 11. If 6 yards of cloth will make 3 coats, whaf^ part of 6 yards will make 1 coat? What part of 6 yards will make 2 coats? 12. What is 1 third of 6 1 What is two thirds o$m& 13. If 3 barrels of cider cost 9 dollars, what par$^| of 9 dollars will 1 barrel cost? What part of 9 doU ^ lars will 2 barrels cost? 14. What is 1 third of 9? What is 2 thirds of 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 one barrel cost? What part of twelve dollars will 3 barrels cost? What i part of twelve dollars will 5 barrels cost? I 18. What is 1 half of twelve 1 What is 3 halves miof 12? \j^^is 5 halves of twelve? ^^^19j^J|Hrels of cider cost twelve dollars, what ^^Hrfrf twelve dollars will 1 barrel cost? What ^^^Hof twelve dollars will 3 barrels cost? What bought for 2 reams of paper, at 5 ^ 19. How much whe^it, at 7 shillings a'L be bought for 2, barrels of cider, at 4 dollars and half a barrel 1 20. How long would it take a man to lay up 10 dollars, if he saves 4 shillings a day 1 21. If a man earn 8 shillings a day, how many dollars would he earn in 10 aays? 22. A man bought twenty pears at thej^Je^f a for 3 cents; how much did they come to ■> 'J 31. A man bought 3 fourths of a^lmhctred weight of yellow ochre for 9 dollars; what was that a hun- dred weight?! % m 32. 9 is three fourths of what number ?f J^, 33. 8 is 4 ninths of what number 1 ^ 34. 9 is 3 tenths of what number 1 35. 10 is 5 sevenths of what numbei T 36. 12 is 3 fifths of what number 1 37. 12 is 4 ninths of what number? 38. 10 is 2 sevenths of what number 1 39. 14 is 7 fifths of what number? 40. 15 is 3 elevenths of what number 1 41. 16 is 2 fifths of what number? 42. 18 is 6 tenths of what number i 43. 20 is 5 ninths of what number 1 44. 21 is 3 ninths of what number 1 45. 24 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 t How much cider at 4 dollars a barrel would it take to pay for a cask of the wine 1 2. 15 is five eighths of how many times 4? 3. If 2 thirds of a pound of coffee cost 18 cents, laow much would a pound cost $ How many oranges at 5 cents apiece, might be bought for a pound? ,.4. JJ8 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 hundred weight come to at the same rate 1 < 6. 20 is 4 sevenths of how many times 6? Sect. 6.] ARITHMETIC. 75 7. A man sold a cow for 21 dollars, which was only seven tenths of what she cost him; how much did she cost him? When he bought her, Le pai for her with cloth at 8 dollars a yard; how many yards of cloth did he give 1 8. 21 is 7 tenths of how many times 81 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 young- est son; what was his age 1 and what was the age of his youngest son? 10. 24 is 3 fifths of how many times 10? 11. 27 is 3 fifths of how many times 7 1 12. 28 is 7 tenths of how many times 9 1 13. 30 is 5 eighths of how many times 7 1 14. 32 is 4 sevenths of how many times 61 15. 36 is 9 eighths of how many times 5? 16. 40 is 8 ninths of how many times 81 17. 42 is 6 fifths of how many times 41 18. 45 is 9 eighths of how many times 61 19. 48 is 8 ninths of how many times 7? 20. 50 is 5 sevenths of how many times 8 t 21. 54 is 9 sixths of how many times 7 1 22. 56 is 7 ninths of how many times 10? 23. 60. is 10 sevenths of how many times 4 T 24. 63 is 9 eighths of how many times 5 I 25. 64 is 8 ninths of how many times 7 1 26. 70 is 10 sevenths of how many times 8 1 27. 72 is 9 fifths of how many times 61 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 l! 2. A boy gave 5 apple^to one of his companion! 76 ARITHMETIC. [Part 1. which was 1 fourth of what he had; how many had he 14 3. A man paid away 4 dollars, which was 2 thirds of all the money he had; how much had he? • 4. A man sold a watch for 18 dollars, which was 3 fourths of 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 1 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 1 8. There is a pole two thirds under water, and 4 feet out; how long is the pole ?. 9. There is a pole two 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 eherries, and 2 sevenths bear peaches, and 10 trees bear plums; how many trees ar^e there in the orchard? and how many of each sort 1 11. There is a school, in which 2 ninths of the boys learn arithmetic, 3 ninths learn grammar, 1 ninth learn geography,"1 ninth learn geometry t and 12 learn to write; how many are there in the school V and how many attending to each study 1 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 1 Sect. 7.] ARITHMETIC. 81 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 of 35? 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 fifths 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? 82 ARITHMETIC. [Part 1. 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 7 v 14. Three times fourteen are how many ?3 15. Three times fifteen are how many? tt 16. Three times sixteen are how many? ■ 17. Three times seventeen are how many? 18. Three times eighteen are how many 1 19. Three times nineteen are how many? 20. Three times twenty are how many? 21. Four times eleven are how many 'L 22. Four times twelve are how many? 23. Four times thirteen are how many 1 24. Four times fourteen are how many 7 25. Four times fifteen are how many 1 26. Four times sixteen are how many 1 27. Four times seventeen are how many 1 28. Four times eighteen are how many? 29. Four times nineteen are how many 1 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 1 35. Five times fifteen are how many £ 36. Five times sixteen are how many 1 37. Five times seventeen are how many 38. Five times eighteen are how many 1 39. Five times nineteen are how many 1 40. Five times twenty are how many? , 41. Six times eleven are how many?: 42. Six times twelve are how many ?*j> Sect. 7.] ARITHMETIC. 43. Six times thirteen are how many 1 J 44. Six times fourteen are how many 1 45. Six times fifteen are how many % 46. Six times sixteen are how many 1 47. Six times seventeen are how many t 48. Six times eighteen are how many 1 49. Six times nineteen are how many? 50. Six times gtwenty are how many 1; 51. Seven times eleven are how many 1 52. Seven times twelve are how many V 53. Seven times thirteen are how many 1 54. Seven times fourteen are how many" 55. Seven times fifteen are how many T 56. Seven times sixteen are how many T 57. Seven times seventeen are how many 58. Seven times eighteen are how many 1 59. Seven times nineteen are how many? 60. Seven times twenty are how many 1 61. Eight times eleven are how many 1 62. Eight times twelve are how many? 63. Eight times thirteen are how many 1 64. Eight times fourteen are how many? 65. Eight times fifteen are how many 1 66. Eight times sixteen are how many 1 67. Eight times seventeen are how many 68. Eight times eighteen are how many I 69. Eight times nineteen are how many? 70. Eight times twenty are how many 1 71. Nine times eleven are how many 1 72. Nine times twelve are how many? 73. Nine times thirteen are how many 1 74. Nine times fourteen are how many 1 75. Nine times fifteen are how many 1 76. Nine times sixteen are how many? 77. Nine times -seventeen are how many 1 78. Nine times eighteen are how many 1 79. Nine times nineteen are how many? 84 ARITHMETIC. [Parti. 80. Nine times twenty are how many T \ 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 1 85. Ten times fifteen are how many? 86. Ten times sixteen are how many 1 < , 87. Ten times seventeen are h#w many 1 88. Ten times eighteen are how many 1 f 89. Ten times nineteen are how many % 90. Ten times twenty are how many 1 ) SECTION VIII. t A. 1. If you cut an apple into two equal parts, what is one of those parts called ?* '«rr 2. How many halves of an apple will* make the whole apple t ]^ 3. If you cut an apple into 3 equal parts, what is 1 of those parts called i^what are 2 of the parts called?; 4. How many thirds of an apple will make the whole apple 1 5. If you cut an apple into 4 equal parts, what is 1 of those parts called t what are 2 of those parts called? what are 3 of them called ?* -r 6. How many fourths of an apple make the whole apple? 7. If an apple be cut into 5 equal parts, what is one of the parts called?** what are 2 of the parts called? what are 3 of the parts called t what are 4 of the parts called? - * Sg€ Section III, article B, remark before question 1 and 17. Sect, ft] ARITHMETIC. 85 8. How many fifths of an apple make the whole apple ? i 9. If an apple be cut into 6 equal parts, what is 1 of the parts called f what are 2 of the parts called? what 31 what 41 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 I of the parts called 1 what are 2 of the parts called? what 3 1 what 4? what 5 1 whaf 61 Leftlie instructer ask the pupil the divisions ofaunit 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 things, Plate II. may be explained and used. It will often be found useful to refer the pupil to the divisions of some sensible object. For the explanation of Plate II see the Key. 12. A man had a bushel of corn and wished to give 1 half of a bushel apiece to some labourers $ how many could he give it to? 13. How many halves are there in 11 14. A man divided 2 barrels of flour among his labourers giving them 1 half of a barrel apiece, how many men did he give it to 1 15 How many halves are there in 21* 16. In 3 bushels of corn how many half bushels 1 17. How many halves are there in 31 18. A boy divided 4 oranges among his compan- ions, giving them 1 half of an orange apiece; how many boys did he give them to? 19. How many halves are there in 41 * Be careful to make the pupil use the plate. He might answer the questions without, but he will not understand their meaning so well. 8 86 ARITHMETIC. [Part 1. 20. A man having some labourers gave them 1 half a dollar apiece; it took 3 dollars and 1 half a dollar to pay them; how many labourers were there? i 21. How many halves are therein 3 and 1 half? 22. How many halves are there in 5? 23. How many halves are there in 7 and I half? 24. How can you tell how many halves there are in any number? Answer. Since there are 2 halves in one, there will be twice as many halves as there are whole ones. p 25. If you had 1 orange, and should divide it among your companions giving them 1 third apiece, how many cduld you give it to? 26. How many thirds are there in 1? 27. If you cut 2 oranges each into 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 in 2 ? in 3 ? in 41 in 5? 31. How can you tell how many thirds there are in any number? Answer. Since there are 3 thirds in one, there will be 3 times as many thirds as there are whole ones. 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, ho;v 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? Sect. &] ARITHMETIC. 89 7. In 6 halves how many times 11 - 8. In 7 halves how many times 1 1 . 9. How can you tell how many whole ones there are in any number of halves 1 10. A man divided some corn among 6 persons, giving them 1 third of a bushel apiece; how many bushels did it take 1 L„ 11. In 6 thirds how many times 1 T ^ 12. In 5 thirds how many times 11 13. A man gave eight paupers 1 third of a dol- lar apiece, how many dollars did it take? 14. In 8 thirds how many times 1 1 15. In 10 thirds how many times 1? 16. How can you tell how many whole ones there are in any number of thirds, 1 17. If a man spends 1 fourth of a dollar in one day, how many dollars will he spend in 8 days 1 How many in 7 days? How many in 11 days 1 18. In 8 fourths how many times 1? 19. In 7 fourths how many times 1? 20. In 11 fourths how many times 11 21. In 13 fourths how many times 1 1 22. In 18 fourths how many times 11 23. How can you tell how many whole ones there are in any number of fourths? 24. If 1 fifth of a barrel of beer will last a fami- ly 1 day, how many barrels will last them 10 days 1 How many 8 days 1 11 days 1 17 days. 1 25. In 10 fifths how many times 1? 26. In 8 fifths how many times 1? 27. In 11 fifths how many times 1? 28. In 17 fifths how many times 1 1 29. In 18 sixths how many times 1? 30. In 23 fifths how many times 1? 31. In 21 sevenths how many times It 32. In 24 eighths how many times 1? 33. In 36 ninths how many times 11 8 * ARITHMETIC. [Part 1. 34. In 30 tenths how many times 1? 1 35. In 35 fourths how many times 11 36. In 37 eighths how many times 11 37. In 43 fifths how many times 1? T 38. In 48 ninths how many times 1 i 39. In 53 tenths how many times 1? 40. In 57 eighths how many times 11 41. In 76 tenths how many times 1? 42. In 78 ninths how many times 11 SECTION IX. A. 1. If a breakfast for 1 man cost 1 third of a dollar; what would a breakfast for two men cost 2. How much is 2 times 1 third T 3. If it take you 1 third of an hour to travel L mile, how long will it take you to travel 3 miles 1 j£ 4. How much is 3 times 1 third TI ***** 5. If 1 man can eat 1 third of a p%md of meat at a meal, how much can 5 men eat? f 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 apiece, how many thirds of a dollar did he give them? how many dollars 1 10. 5 times 2 thirds are how many thirds? how many times 1? 11. If you give 3 men 1 fourth of a dollar apiece, how many fourths of a dollar will it take? 12. 3 times 1 fourth are how many fourths 1 13. If you give 3 men 3 fourths of a bushel of Sect. 9.] ARITHMETIC. 91 corn apiece, how many fourths of a bushel will it take 1 how many bushels 14. 5 times 3 fourths are how many fourths 1 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. J^imes 1 fifth are how many fifths 1 17. flPl 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 flour in a week, how much would they consume in 5 weeks? 20. 6 times 2 sevenths are how many sevenths 1 how many times 11 21. 5 times 3 eighths are how many eighths 1 how many times 1 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 1 27. How much is 5 times 7 tenths 1 28. How much is 6 times 7 eighths 1 29. How much is 9 times 5 eighths? 30. How much is 8 times 5 sevenths 1 31. How much is 7 times 5 sixths? 32. How much is 8 times 7 fourths 1 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 dollar and 1 half, what will two bushels cost? 2. How much is 2 times 1 and 1 half ?* * This is to be understood 2 times 1 and 2 times 1 half, and to be answered thus: 2 times 1 are 2, and 2 times 1 half are 2 halves or 1, which, added to 2, makes 3. ARITHMETIC. [Part 1. 3. If a barrel of cider cost 2 dollars and a half, what will 3 barrels cost 1 4. How much is 4 times 2 and 1 half? 5. If a barrel of beer cost 3 dollars and a half, what will 2 barrels cost 1 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 Pthird of a dollar, what will 3 boxes cost 1 9. How much is 4 times 2 and 1 third? 10. If you give to two persons 3 bushels and 1 third of a bushel of wheat apiece, how many bushels will it take 1 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 1 13. How much is 5 times 2 and 1 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 1 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 consume in 3 months 1 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 1 fourth? 20. How much is 3 times 3 and 3 fourths? 21. How much is 3 times 5 and 1 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? 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 3 fifths? Sect. 10.] 95 ARITHMETIC. 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 71 7. What is 1 half of 9 1 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 one 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 twic as much as one third of one thing. One third of one s 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 orange. Hence one third of four oranges is four thirds 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? * Divide each orange into three parts, and then take one part from each. 96 ARITHMETIC. [Part 1. 19. What is one third of 2 ? of 4? 30. If 3 gallons of wine cost 5 dollars, what is that a gallon? 21. What is 1 third of 51 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 a piece? What would they have apiece if 2 bushels were divided ampng them? What if 3 bush- els? 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 he divided among 5 men, how much will each have? 4 barrels were divided, what would each have T 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 ? of2? of 3? of 4? of 5? of 7? of 8? of9? 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 I? of 2 ? of 3? of 4? of5? of6? of7? of8? of9? of 10? 31. What is 1 ninth of 1 ? of 2? of 3? of 4? of5? of 6? of 7 ? of 8? of9? of 10? of 11? 32. What is 1 tenth of 1? of 2 ? of 3? of 4? of 5? of6? of7? of8? of9? of 10? of 11? of 12? of 13? 33. If 3 yards of cloth cost 2 dollars, what will 1 yard cost? What will 2 yards cost? 34. If 1 bushel of wheat cost 2 dollars, what Sect. 12.J ARITHMETIC. 107^^ under the line is called the denominator, because it gives name to the fraction; and the number above the line is called the numerator, because it sho^ viie number of parts used. Thus T\, 10 is the denominator and 3 the numerator. N. B. The pupil must be made familiar with this mode of expressing fractions, and must be able to apply it to any familiar objects; as apples, oran- ges, &c.; and to the table, before he is allowed to proceed any farther. Particular care must be taken to make him understand what the denominator sig- nifies, and what the numerator, as explained above. The denominator should always be explained first. The following examples are a recapitulation of some of the foregoing sections, for the purpose of showing the application of the above method of writing fractions. See Section VIII. A. A. 1. In 2 how many times 11 Ans. f * 2. In 3 how many times | 1 Ans. $. 3. In 2 how many times £? Ans. f. 4. In 4 how many times ^ 1 5. In 6 how many times \ 1 6. In 7 how many times } 1 7. In 8 how many times \ 1 8. In 2£t how many times \ 1 9. In 3£ how many times \ 1 10. Reduce 4i to an improper fraction-! 11. Reduce 3f to an improper fraction. 12. Reduce 5 J to an improper fraction. * When the numerator is larger than the denominator, the fraction i called an improper* fraction. 12 1-2 is read two and 1 half. It is called a mixed number. hat is, to find how many fifths there are in four and 1 fifth. 108 ARITHMETIC. [Part 1. 13. Reduce 6J to an improper fraction. 14. Reduce 8T\ to an improper fraction. 15. Reduce 9f to an improper fraction. B. 1. |- are how many times 1? 2. | are how many times 1? 3. \ are how many times 11 4. | are how many times 1? 5. f are how many times 1? 6. y are how many times I?' 7. y are now many times 1? 8. y are how many times 1? 9. y are how many times 1? 10. \£ are how many times 1? jSfee Section IX. A. 1. How much is 3 times £ 1 2. How much is 4 times i? 3. How much is 3 times f 1 4. How much is 4 times ^? 5. How much is 5 times f 1 6. How much is 6 times J? 7. How much is 8 times T3y? 8. How much is 9 times % 1 9. How much is 10 times f? 10. How much is 9 times f 1 B. 1. How much is 3 times 2J1 2. How much is 4 times 3|? 3. How much is 5 times 6|? 4. How much is 6 times 4|? 5. How much is 7 times 5f? G. How much is S times 6f t 7. How much is 4 times 10^ 1 Sect. 12.] ARITHMETIC. 109 8. How much is 9 times 7|? 9. How much is 8 times 9$? 10. How much is 10 times 7f 1 Sfec Sections V. & X. 1. What is £ of 6 1 3. What is J of 8? 5. What is | of 9? 7. What is \ of 14 2 9. What is f of 5? 11. What is | of 7? 13. What is \ of 17? 15. What is £ of 27? 17. What is $ of 47? 19. What is 4 of 65? 2. What is £ of 6? 4. What is | of 9? 6. Whafisjof 101 8. What is | of 5? 10. What is £ of 7? 12. What is f of 35? 14. What is f of 26? 16. What is £ of 37? 18. What is | of 42? 20. What is \ of 75 ) See Sections VI. & XI. A. 1. 2 is i of what number? 2. 4 is £ of what number? 3. 8 is | of what number? 4. 1^ is - of what number? 5. 2 J is £ of what number? 6. 4f is i of what number? 7. 6f is ^ of what number? 8. 7f is i of what number? 9. 8J is | of what number? 10. 9T\ is TV of what number? B. 1. 4 is § of what number? 2. 6 is f of what number? 3. 8 is of what number? \ 10 110 ARITHMETIC. [Part 1. 4. 12 is % of what number? 5. 15 is f of what number 1 6. 18 is f of what number? 7. 20 is | of what number 1 8. 24 is | of what number? 9. 28 is | of what number? 10. 30 is T57 of what number 1 11. .3 is f of what number? 12. 4 is f of what number? 13. 5 is 4 of wjiat number? 14. 8 is | of what number? 15. 9 is f of what number? 16. 17 is | of what number? 17. 25 is £ of what number 1 18. 38 is * of what number? 19. 43 is f what number? 20. 54 is V° of what number 1 Miscellaneous Examples. 1. A man sold 8 yards of cloth for 3£ dollars a yard; what did it come to? 2. A man sold a horse for 76 dollars, which was | of what it cost him; how much did it cost him? 3. A man spld J of a gallon of wine for 40 cents; what was that a gallon? 4. If it will take If yards of cloth to make a coat, how many yards will it take to make 7 coats? 5. If 1 horse consume 3i bushels of oats in 2* days, how much would 2 horses in 5 days? 6. If when the days are 9£ 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 7j dollars a yard, and received 8 firkins of butter at 6| dollars a firkin; how much was then due to him 1 8. Two men are 38 miles apart, and are travel- Sect. 13.] ARITHMETIC. Ill ing towards 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 1 How much in 2 hours 1 In how many hours will they meet 1 At what distance from each place from which they set out? SECTION XIII. A. 1. If you give \ of an orange to one hoy, and \ to another, how much more do you give the first, than the second 1 2. £ of an orange is how many £ of an orange 1 3. If you give \ of an orange to one boy, and % to another, how many \ would you give away t How many a would you have left 1 4. £ and £ are how many {1 5. A man gave to one labourer £ of a bushel of wheat, and f to another; how many | of a bushel did he give to both 1 How many bushels 1 6. ^ and J are how many £1 How many times 11 7. A man gave \ of a barrel of flour to one man, and l of a barrel to another; to which did he give the most 1 8. \ is how many \ 1 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 1 10. i Is how many £ 1 11. A man bought f of a yard of cloth at one time, and f of a yard at another; at which time did he buy the most 1 12. | are how many } 1 13. A man wished to give J of a bushel of whea*. 112 ARITHMETIC. [Parti. to one man, and { of a bushel to another; but he could not tell how to divide it. Another man stand- ing by advised him to divide the whole bushel into six equal parts first, and then take £ 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 1 14. | is how many £ 1 \ is how many 11 { and i are how many } 1 15. A man paying some money to his labourers, gave each man ^ of a dollar, and each boy }of a dollar; how much more did he give to a man than to a boy? 16. What is the difference between £ and |? 17. If a man can earn £ 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 £ and \ 1 19. A boy distributing some nuts among his com- panions, gave i of a quart to one, and \ 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 £ 1 21. A man having two bushels of grain to distri- bute among his labourers, wished to give \ of a bushel to one, and § 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 tot equal parts, or sixths, and then to distribute those parts. How many sixths did he give to each? 22. | is how many \? 23. A man had a horse, and a cow, and a sheep. The horse would eat f of a load of hay in the win- ter, the cow and the sheep \. How many \ of a load would each eat 1 How many } would they all eat! How many loads 1 Sect. 13.] ARITHMETIC. 113 24. A boy having a quart of nuts, wished to di- vide them, so as to give one companion another J, and a third \ of them; but in order to make a proper division, he first divided the whole into eight equal parts, and then he was able to divide them as he wished. How many eighths did he give to each 1 How many eighths had he left for himself? 25. i is how many \1 £ is how many }1 ^ and £ and | are how many 11 26. A man gave J of a barrel of flour to one man, and £ of a barrel to another; to which did he give the most 1 How much 1 27. Which is the largest £ or f 1 How much the largest? 28. A boy having a pound of almonds, said he intended to give \ of them to his sister, and \ to his brother, and the rest to his mamma. His mam- ma smiling said she did not think he could divide them so. O 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 1 29. \ is how many T\ 1 } is how many T*y 1 and } are now many T^? 30. Mr. Goodman having a pound of raisins, said he would give Sarah 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 J and £ of them. How many twelfths would each have? 31. £ and J and } are how many Tly 1 32. George bought a pine apple, and said he would give | of it to his papa, and § to his mamma, and 737 to. his brother James, if he could divide it. James took it, and cut it into twenty equal pieces, 10 *; 114 ARITHMETIC. [Part 1. and then distributed them as George had desired. How many twentieths did he give to each 1 33. i is how many fa 1 | is how many 2V • f *s how many fa 1 fa is how many fa? 34. i is how many fa 1 35. £ is how many fa? 36. £ is how many £? 37. £ is how many T^ 7 38. I are how many £ 7 39. § are how many ^T 7 40. { is how many fa 7 41. I are how many x!j 7 42. £ are how many fa 7 43. 4 are how many fa ^ 44. $ are how many fa 7 45. f are how many fa 7 46. I are how many 3^ 7 47. T\ are how many fa 7 48. Reduce \ to sixths and ^ to sixths. 49. I and f are how many £ 7 50. Reduce £ and \ to eighths. 51. i and \ are how many \ 7 52. i and \ are how many 7 53. § and £ are how many \ 7 54. I and f are how many | 7 55. J and £ are how many £ 7 56. I and § are how many fa 1 57. £ and a and j are how many | 7 58. i and f and fa are how many TV t 59. I and J are how many fa 7 60. § and £ and j are how many fa 7 61. J and T\ and J are how many fa 7 62. | and £ and f and £ and TV are how many -^7 63. i and f are how many TV? 64. § and J are how many fa 7 65. § and f are how many 7 §(5. I less £ are how many £ 7 Sect. 13] ARITHMETIC. 115 67. f and £ less ^ are now many^ t 68. £ less ^ are how many? 69. J less f are how many ^? 70. £ less f are how many ^T1 71. \ and £, and and f^, less J are how many 72. | and and f, and y1^, and les9 £ are how many ^ 1 73. 4 and # are how many? 74. £ and f are how many ?y? 75. 4 and £ are how many -fa 1 When the denominators in two or more frac- tions are the same, the fractions are said to have a common denominator. Thus f and f have a com- mon denominator. 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 subtract the numerators and write their sura 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 § and £ to a common denominator. Note. They may be reduced to twelfths. If it cannot be immediately oeen what number must be the common denominator, it may be found by multiplying all the denominators together; fo*? that will always produce a number, divisible by all* the denominators. ,77. Reduce J and f to a common denominator. 78. Reduce § and £ and £ to a common denomi- nator. 79. Reduce \ and f to a common denominator. 80. Reduce £ and f to a common denominator. 116 ARITHMETIC. [Part 1. 81. Reduce i and § and J to a common denomi- nator/ 82. # Add together § and f. 83/ Add together % and T5T. 84. Add together | and T\. 85. Add together £ and £ and 86. Subtract £ from 87. Subtract ^ from f. 88. Subtract f from §. 89. Subtract f from |. B. 1. Mr. F. said he would give £ of a pine ap- ple to Fanny, and § 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 himself. Could you have told if you had been there 1 How would you divide it 1 How much would be left 1 2. A man sold \\ bushels of wheat to one man, 4f bushels to another; how many bushels did he sell to both 1 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 6f yards of another kind; how many yards in the whole 1 5. A man bought f of a barrel of beer at one time, 2^ barrels at another, and 6| at another; how much did he buy in the whole 1 6. A man bought one sheep for 4| dollars, and another for 5f dollars; how much did he give for both? 7. There is a pole standing, so that £ of it is in the mud, and \ of it in the water, and the rest out of the water; how much of it was out of the 4rater? Sect. 13.] ARITHMETIC. fc . 117 9 8. A man having undertaken to ao a piece of work, did £ of it the first day, £ 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 | 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 1 C. It will be seen by looking on plate III, that f is the same as ^, and that £ is the same as \, and that | is the same as ~ ; f, f, can therefore be re- duced to ^, and $- to §. This is called reducing fractions to their lowest terms. 1. Reduce f to its lowest terms.* Arts, f. 2. Reduce T5F to its lowest terms. 3. Reduce £ to its lowest terms. 4. Reduce T\ to its lowest terms. 5. Reduce \% to its lowest terms. . 6. Reduce -fa to its lowest terms. 7. Reduce to its lowest terms. 8. Reduce to its lowest terms. 9. Reduce £f to its lowest terms. 10. Reduce T\ to its lowest terms. 11. Reduce to its lowest terms. 12. Reduce to *ts lowest terms. 13. Reduce to its lowest terms. 14. Reduce $f to its lowest terms. Note. It will be seen by the above section that if both the numerator and denominator be multi- plied 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 frac- tion will not be altered. * If this article should be found too difficult for the pupil, he may omit it til! after the next section. 118 _ — ARITHMETIC. [Parti. SECTION XIV. A. 1. A boy having £ of an orange gave away £ of that, what part of the whole orange did he give away? 2. What is j. of ± 1 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 \ of £ 1 5. A boy had ^ 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 j. 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 1 8. What is | of i? |j. A man having | a barrel of flour, sold £ of tha|; how much did he sell 1 fO. What is i of 11. If an orange be cut into 4 equal parts, and each of those parts be cut into 3 equal parts, how many parte will the whole orange be cut into? 12. 'What is \ of ±? 13. A boy having £ of a quart of chestnuts, gave away \ of what he had. What part of the whole quart did he give away 1 14. What is \ of£? 15. What is i of }? 16. A man owning \ of a ship, sold % of his share; what part of the ship did he sell, and what part did he then own 1 17. What is i of |? 18. What is £ of Sect. 14.] ARITHMETIC- 119 19. What is i of 11 20. What is | of |? 21. What is ^ of 4? 22. What is } of £? 23. What is } of} 1 24. What is | of |? 25. What is £ of }! 26. What is | of |? 27. What is a of |? 28. What is 4 of |? 29. A boy having § of an orange, (that is, 2 pieces,) gave his.sister \ of what he had; how many thirds did he give her? 30. What is \ of f? 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 £ or three pieces, if you cut them all in two, you can give me £ of them. But | of \ is therefore I shall have, f of the whole pine apple. 32. What is % of |? 33. A man owning f of a share in the Boston bank, sold i of his part. What part of a share did he sell? 34. What is | off? 35. A man owning £ of a ship, sold | of his share; what part of the whole ship did he sell 1 What part had he left? 36. What is £off? 37. Whatis£off? 38. What is £ off? 39. What is | off? 40. What is i of y? 41. What is \ of f? 42. A man owning f of a share in the Boston bank, sold \ of his part; what part of a whole share did he sell? 120 ARITHMETIC. [Parti. 43. What is i of f? 44. What is -} off? 45. A boy having f of a water melon, wished to divide his part equally between his sister, his bro- ther, 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 wasr each piece? 46. What is i of f? 47. What is} of |? 48. What is |off? 49. What is £ off? 50. WhatisTVof^? 51. What is i of i? 52. What is | of i? 53. What is i of |? 54. What is £ off? 55. What is i of |? 56. What is § of f1 57. What is i off? 58. What is | off? 59. What is f off? 60. What is f off? 61. What is f of f? 62. What is £ of f? 63. What of |? 64. What is T'T off? 65. What is T\ of f 1 66. What is f of r\l 67. What is J off? 68. What is $ of f? 69. If a yard of cloth cost 2| dollars, what will } of a yard cost? / 70. What is \ of 21? 71. A boy had 21 oranges, and wished to give of them to his sister, and i to his brother, but In Sect. 14.] ARITHMETIC. 121 did not know how to divide them equally. His bro- ther 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 £ of 2-L? 73. A man bought 4 bushels of corn for 3| dol- lars; what part of a dollar did 1 bushel cost? Change the 3| to thirds, and then find \ of y as above. 74. What is | of off 75. If 5 bushels of wheat cost 7£ dollars, what is that a bushel %» 76. What is i of 7f? 77. A man bought 6 gallons of brandy for 3| dol- lars; what was that a gallon? 78. What is } of 8f? 79. A man bought 7 gallons of wine for 8^ dol- lars; how much was that a gallon? 80. What is-i of 84? 81. A man bought 10 pieces of nankin for 6| dollars; how much was it a piece 1 82. What is TV of 6ft 83. If 9 bushels of rye cost 7$ dollars, what is that a bushel 1 84. What is£ of 7f? 85. What ib | of 5|? 86. What is £ of 8£? 87. What is i of 6^? , 88. What is | of 9|? 89. A man bought 7 yards of cloth for 18f dol- lars; what was that a yard ? What would 3 yards cost at that rate? * 90. What is | of 18f? What is j of 18£ f 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 i of 27f 1 What is } of 27|? 122 ARITHMETIC. [Part 1. 93. If 6 barrels of flour cost 38f dollars, wha^ would 10 barrels cost at that rate? 94. What is y of 38f? B. 1. A man bought a piece of cloth for 42£ dol- lars, and was obliged to sell it for f 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 4| 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 1 5. There is a pole standing so that -| of it is in the water, and § as much in the mud; how much is in the mud 1 6. If a man can travel 13f miles in 3 hours, how many miles will he travel in 8 hours 7 7. If 5 horses will eat 26J loads of hay in a year, what will 8 horses eat in the same time? 8. If 4 cocks' will empty a cistern in 6f hours, how long will it take 7 cocks of the same size to 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 t 2. How many times £ are there in 21 How many times | are there in 21 Sect. 15.] ARITHMETIC. 3. A man having 3 bushels of corn distributed it among some poor persons, giving them J of a bush- el each; to how many did he give it 1 Note. Find jirst how many he would have given it to, if he had given £ of a bushel to each. 4. In 3 are how many times £1 How many times £ 1 5. If | of a barrel of flour will last a family one month, how long will 4 barrels last the same family t How long will 6 barrels last 1 How long will 10 barrels last 1 6. How many times i3 § contained in 4 1 How many times in 0? How many timesjn 10? 7. Iff of a bushel of wheat will last a family one week, how many weeks will 6f bushels last the same family? 8. How many times is £ contained in 6| 1 9. There is a cistern having a cock which will fill it in § of an hour; how many times would the cock fill the cistern in 3| hours 1 10. How many times is § contained in 3| 1 11. How much cloth at 1£ dollars (that is f dol- lars) a yard can be bought for 4 dollars 1 12. How many times is \\ or f contained in 4? 13. A man distributed 8£ bushels of wheat among some poor persons, giving l£ bushels to each ; how many did he give it to 1 14. How many times is \\ contained in 8£? 15. If a soldier is allowed l£ pounds (that is £ of a pound) of meat in a day, to how many soldiers would 6| pounds be allowed? 16. How many times is l£ contained in 6| 1 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 2i dollars a box, how many boxes of rai- sins can be bought for 10 dollars? 20. How many times is 2^ contained in 10 1 124 ARITHMETIC. [Part 1. 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 9|? 23. At 1| dollars a barrel, how many barrels of raisins can be bought for 9^ dollars? 24. How many times is 1| contained in 9|? 25. At -J of a dollar apiece, how many pieces of nankin can be bought for 8f dollars? 26. How many times is f contained in 8f 1 27. At f of a dollar a pound, how many pounds of tea can be bought for 7§ dollars? 28. How majay times is £ contained in 7§? 29. How many times is 3£ contained in 7§? 30. How many times is 5£ contained in 17 1 31. How many times is 4| contained in 9| 1 32. How many times is 3^ contained in 12* 1 B. 1. At TVof 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 TV contained in £? 3. A man having £ of a barrel of flour distributed it among some poor persons, giving them } of a barrel apiece; how many did he give it to? Note. Change both fractions to twelfths; that ?s, reduce them to a common denominator. 4. How many times is £ contained in f? 5. If a pound of almonds cost \ of a dollar, how many pounds can be bought for f of a dollar T Note. Reduce the fractions to a common denomi- nator. 6. How many times is | contained in f? 7. If a piece of nankin cost f of a dollar, how many pieces can be bought for 4f dollars 1 that is V dollars 1 8. How many times is § contained in 4f-? 9. If a bushel of barley cost J of a dollar, how Part I.] ARITHMETIC. 125 %any bushels can be bought for f of a dollar 1 How many for If dollars 7 10. How many times is f contained in J 1 How many times in If 1 11. How many times is ^ contained in f 1 12. How many times is $ contained in 11 TABLES OF COINS, WEIGHTS, AND MEASURES. Table L—Federal Money. 10 mills 10 cents M * 10 dimes % 10 dollars make 1 cent. 1 dime. 1 dollar. 1 eagle. Table JJ.—Sterling Monet. 4 farthings q. 12 pence 20 shillings 6 shillings 28 shillings make 1 penny. 1 shilling. 1 pound. 1 dollar. 1 guinea. d. s. Table HX—Troy Weight. 24 grains gr. 20 pennyweights 12 ounces make 1 pennyweight, dwt, "1 ounce. ;| ^ oz. "1 pound. "tib. 16 drams dr. 16 ounces 28 pounds Table TV.—Avoirdupois Weight. make 1 ounce. 1 pound, t 1 quarter of a ( hundred weight. oz. Ib. 11 126 [Part 1. ARITHMETIC. 4 quarters make 1 hundred weight. 20 hundred weight" 1 ton. * Table V.—Cloth Measure. 2£ inches make 1 nail. 4 nails " 1 quarter of a yard. 4 quarters 44 1 yard. 3 quarters " 1 ell Flemish. 5 quarters " 1 ell English. 6 quarters " 1 aune ot ell French. Table VI—Wine Measure. 1 gallon. 1 barrel. 1 hogshead. 1 pipe. 1 tun. Table VII.—Dry Measure. 2 pints make 1 quart. 8 quarts" 1 peck. 4 pecks " 1 bushel. Table VIII.—Measure of Time. 60 seconds sec. 60 minutes 24 hours 7 days 4 weeks 13 months 1 day and i 6 hours, or 365 > days and 6' hours J 4 gills make 2 pints <( 4 quarts *\ 31£ gallons u 63 gallons K 2 hogshead <« 2 pines T. qr. 7- pt. qt. gal. bar. hd. qt. pk. bu. make 1 minute. min. (c 1 hour. h. ii lday. • d. «( 1 week. w. «< 1 month. m. 1 year. yr. Parti.] ARITHMETIC. 1ST 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, be- cause they are the months used in calendars. The names of the months, and the number of days in each, are as follows: Number of days, 31 28, in leap year 29. 31 30 31 30 31 31 30 31 * 30 31 Miscellaneous Examples. 1. In 2 pounds how many ounces? 2. In 8 yards how many quarters 1 3. In 3 quarters of a yard how many nails 1 4. -fa of a dollar is how many cents 1 5. How many farthings is f of a penny? 6. How many pence is -J- of a shilling? 7. J of a yard is how many quarters and nailg? Names. 1. January 2. February 3. March 4. April 5. May 6. June 1 { 7. July 8. August September October November TO { { I , whose price and quantity are given, and then to find how much of the other article that money will buy. 152 [Part 2. KEY. 6. If 2 apples cost 4 cents, 1 will cost 2 cents, and 4 will cost 8 cents. Or 4 apples will cost 2 times as much as 2 apples. 22. Find how many times|2 pears are contained in 20 pears, which is 10 times. 10 times 3 cents are 30 cents. Or, first find what 20 pears would come to, at 3 cents apiece; and since it is 2 for 3 cents, instead of 1 for 3 cents, the price will be half as much. 23. See how many times you can have 5 cents in 30 cents, and you can buy so many times 3 eggs. 30 is 6 times 5, and 6 times 3 are 18. 18 eggs. 24. 10 dollars a week, and 40 dollars a month. 25. 5 dollars are 30 shillings, which is 10 shil- lings a day. 26. 5 dollars apiece. SECTION V. In this section the principle of fractions is applied to larger numbers, but such as are divisible into the parts proposed to be taken. The pupil, whn is fa- miliar with what precedes, will easily understand the examples in this section. They require nothing but division and multiplication. A. Let the pupil explain each example in the following manner. What is 1 sixth of 18 1 Ans.3. Why? Because 6 times 3 are 18; therefore if you divide 18 into 6 equal parts, one of the parts will be 3. To find this answer on the plate; on the 6th row, the pupil will find 3 times 6 make 18; this will direct him to the third row, where he will find 6 Sect. 5.] 153 KEY. times 3 are 18. Consequently, he will see 18 divid- ed into 6 equal parts. It will be well to let the pu- pil prove a large number of the examples on the plate. The pupil will be very likely to say 3 is the 6th part of 18, because 3 times 6 are 18. Be careful to make him say it the other way, viz. 6 times 3 are 18. 14. 1 third of 9 is 3; | is 2 times as much as |; therefore f of 9 is 6. 19. 1 barrel will cost £ part of 12 dollars; 3 bar- rels will cost § of 12 dollars. 7 barrels will cost J of 12 dollars. 37. What is J of 32? | of 32 is 4, f are 5 times 4, or 20. B. 11. } of 20 is 4; i are 7 times 4, or 28; and 28 is 4 times 6, and £ of 6. C. 3. 1 half of 10 is 5, J of 10 are 4; 5 and 4 are 9. He gave away 9 and had 1 left. 4. 1 .yard will cost £ of what 3 yards cost. J- of 6 dollars is 2 dollars. 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 6. 3 apples will cost £ of what 9 cost; or 6 cents. 7. 2 is | of 3; therefore 2 oranges will cost § of what 3 cost. | of 18 cents are 12 cents. 8. f of 25 are 20. The 10 apples cost 20 cents, which was 2 cents apiece. 11. $ of 42 are 12, and 6 times 12 are 72. 72 dollars. 13." 3 is £ of 4. £ of 12 dollars are 9 dollars. Or 4 yards at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards. 14. Solved like the 13th. Ans. 15 cents. 15. Since 1 is £ of 3, 7 is £ of 3. \ of 15 cents are 35 ce%ts. Or, 3 oranges at 15 cents, is 5 cents apiece: 7 times 5 are 35 cents. 154 [Part % KEY. Note. In questions of this kind it is generally the simplest way to find what 1 article will cost, then it. may easily be told how much any numoer will cost. 19. 4 men would do it in 1 half the time that 2 would do it. Or, you may say, if 2 men would do it in 6 days, 1 man would do it in 12 days, and 4 men in | of that time, or 3 days. SECTION VI. A. 4. 2 halves of any number make the whole number. Therefore 2 is 1 half of 2 times 2; or 4. It is \ of primes 2, or 8. Let the pupil answer these questions in the fol- lowing manner: 4 is £ of 3 times 4; 3 times 4 are 12. 5 is 4 of 7 times 5; 7 times 5 are 35. , B. 2. 4 is 2 times 2. 4. 6 is 2 times 3. 16. 2 thirds of any number is twice as much as £ of the same number. If 4 is § of some number, then 1 half of 4 or 2 is £ of that number ; 2 is £ of 6: therefore 4 is § of 6. 20. If 6 is | of a number, £ of 6 or 2 is j of the same number; 2 is \ of 8; therefore 6 is J of 8. 23. It is evident that | of a pound will cost only \ of what f will cost; If cost 6 cents, | will cost 2 cents, and the whole pound 14 cents. 26. It will probably be perceived by this time, that f of a lumber being given, it is necessary to find 4? and then the number is easily found; 4 be- ing f, 2 is j, and 2 is a of 14. * 45. 24 being |, | of 24 or 3 will be }; 3 is £ of 27, Sect. 6.] 155 KEY. C. 6. 20 being4,5is and 5 is| of 35; and35 is 5 times 6, and f of 6. D. 4. 18 is 3 times 6, and 6 is | of 4 times 6, or 24. Ans. 24 dollars. 6. 54 is f of 48; 12 yards at 48 dollars is 4 dol- lars ia yard. He gained 6 dollars. 7. 10 feet is § of 15 feet. 8. If f are under water, there must be £ out of the water. 4 is | of 12. 9. If | are under water there must be f out of the water. 6 is £ of 10. 10. 4 and $ are 4. 4 bear cherries and peaches; consequently, the 10 which bear plums must be the other f; 10 is f of 35. 10 bear peaches and 15 bear cherries. 11. f, and and and |, are %; therefore 12 must be the other f of the whole* The whole number is 54. Miscellaneous Examples. 6. The grey-hound gains upon the fox 4 rods in a minute. It will take him 20 minutes to gain 80 rods. 8. of 24. Or you may say, 1 sheep would cost 3 dollars, and 3 sheep 9 dollars. 9. 30 horses will eat 10 times as much as 3 horses. 11. 10 dollars apiece, and 2 dollars a yard. 12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for 5 weeks. 14. It would take them 5 times as long to eat 40 bushels, as it would to eat 8 bushels. 15. 4 horses would eat 4 bushels in 3 days, and it would take them 9 times as long to eat 30 bush- els, mns. 27 days. 156 [Part % KEY. 16. If 2 men spend 12 dollars in 1 week, 1 man will spend 6 dollars in 1 week, and 30 dollars in 5 weeks, and 3 men would spend 3 times as much, or 90 dollars. 17. The shadow of the staff is § of the length of the staff, therefore the shadow of the pole must be | the length of the pole. 18 feet is § of 27 feet. 20. It would take 2 men 3 times as long to do it as it would 6 men. 23. 8 men would do a piece of work 1 half as large in 2 days, and it would take 2 men 4 times as long to do it, or 8 days. 28* He must sell it for 56 dollars in order to gain 16 dollars. 56 dollars is 7 dollars per barrel. 29. It cost him 35 dollars, and he must sell it for 45 to gain 10 dollars; 45 dollars is 9 dollars a firkin. 30. Ans. 56 cents, see section VI. 33. If it would last 3 men 10 months, it would last 1 man 30 months, and 5 men 6 months. 34. There are 8 times 5 in 40, and since the other would build as many times 9, as the first does 5, he would build 8 times 9 or 72 rods SECTION VII. A. 13. f of 20 is 4, | are 16; 16 being f 2 is $; 2 is | of 14, and 16 is f of 14. 16. f of 28 are 12; 12 is 2 times 6, and 6 is } of 48, (12 is | of 48) and 48 is 6 times 7 and f of 7. B. 1. | of 15 are 12; 12 is 6 times 2; 2 is TV of 20 (12 is T80- of 20); i of 21 is 7; 20 is %times 7 and 4 of 7. Sect, a] KEY. 2. | of 18 are 24; 24 is | of 27; 4 of 35 is 5; 27 is 5 times 5 and f of & C. This article contains the multiplication table, in which the numbers from 10 to 20 are multiplied by the ten first numbers. SECTION VIII. Explanation of Plate II. Plate I, which has been used in the preceding sections, presents each unit as a simple object and undivided. Plate II, presents the units as divisible objects, the different fractions of which form parts, and sums of parts of unity. This plate is divided into ten rows of equal squares, and each row into ten squares. The first row is composed of ten empty squares, which are to be represented ■ > the pupil as entire units. The second row presents ten squares, each divided into two equal parts by a vertical line, each of these parts of course represents one half. In the third row, each square is divided into three equal parts, by two vertical lines, each part repre- senting one third, Spc. to the tenth row, which is di- vided into ten equal parts, each part representing one tenth of unity. N. B. In plates II and III, the spaces and not the marks are to be counted. Be careful to make the pupil understand, 1st, that each square on the plate is to be considered as 14 158 [Part.% KEY. an entire unit, or whole one. 2d, explain the divi- sions into two, three, four, &c. parts. 3d, teach him to name the different parts. Make him observe that the name shows into how many parts one is divided and how many parts are taken, in the same manne as it does when applied to larger numbers. -f f°r example, shows that one thing is to be divided into 7 equal parts, and 4 of those parts are to be taken. 4th, make the pupil compare the different parts to- gether, and observe which is* the largest. Ask him such questions as the following: Which are the smallest halves or thirds? Ans. Thirds. Why? Because, the more parts a thing is divided into, the smaller the parts must be. A. 15. On plate II., count two squares in the sec- ond row, and then ascertain the number of spaces or halves in them. There are 4 halves. 21. In the 2d row take 3 squares and 1 space in the 4th square; then count the spaces. Ans. 7 halves. 37. In the 3d row take 5 squares, and 2 spaces in the 6th; then count the spaces or thirds. Ans. 17 thirds. 54. In the 5th row take 6 squares, and 4 spaces in the 7th square; then count the spaces or fifths. Ans. 34 fifths. B. 2. This operation is the reverse of the last. In the 2d row count 4 spaces or halves, and see how many squares or whole ones it takes. It will take 2. 38. In the 9th row count 48 spaces or 9ths, and see how many squares or whole ones it takes. It will take 5 squares and 3 spaces in the 6th. Ans. 5 whole ones and f. KEY. 169 SECTION IX. A. 2. \ signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times I third is 2 parts, or |. 6. 7 times \ is f, or 2£. 10. On the plate in the 3d row, 5 times § are V°» which takes 3 squares and 1 space. Ans. 3£. 24. In the 9th row take 4 spaces or 9ths, and re- peat them 5 times, which will make \9, and will take 2 squares and 2 spaces. Ans. 2f. B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 2, which added to 8 make 10. 18. 4 times 3 are 12, and 4 times f are or three whole ones, which added to 12 make 15. 32. 2 times 3 are 6, and 2 times ^ are $, which added to 6 make 6-f. 40. 10 barrels of cider at 3 dollars and £ a bar- rel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times £ is 5/, or 8 and f of a dollar. Ans. 38f dollars. C. 2. £ to each would be 3 times £, or f, which are 2j oranges. 3. V or 2 bushels. 4. 7 times J are y, or gallons. 5. 8 yards and f or 2 yards, that is, 10 yards. 6. 4 times 2 are 8, and 4 times f are !/, or 2|, which added to 8 make 10f bushels. 12. It would take 1 man 3 times as long as it would 3 men. Ans. 13| days. 14. 3 men would build 3 times as much as 1 man; and in 4 days they would build 4 times as much as in 1 day. Ans. 38f rods. 15. Ans. 12 yards. 160 [Pari & KEY, SECTION X. A. 21. £ of 1 is |. I of 2 is 2 times as much, or f. £ of 4 is or 1| i of 5 is |, or If. f of 6 is f, or 2. iof7isf,or2£. 27. i of 1 is ±. | of 2 is §. } of 3 is f. \ of 7 is |, or If. This manner of reasoning may be applied to any number. To find j of 38: it is y, for j of 38 is 38 times as much as | of 1, and \ of 1 is |, consequent- ly | of 38 is \8, and 3T8 is 5f 40. To find | of a number, \ must be found first, and then § will be 2 times as much. £ of 7 is J, and 2 times f are y, or 4f. 74. £ of 50 is S9% or 5f ; % is 4 times as much; 4 times 5 are 20, 4 times f are V» or which added to 20 make 22§. Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Arts. If apiece. 3. J of 3 is |; | of a bushel apiece. 4. J of 7 is 4J; he gave away 4£ and kept 2f. 6. 1 half dollar a yard, or 50 cents. 7. i of 7 is |, or If; | of a dollar is f of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. £ of 8 is If. f of 100 is 33f. Ans. 1 dollar and 33f cents, or it is 1 dollar and 2 shillings. 9. iff 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and §, and 2 bushels will cost 5£ dollars, Ans. 5 dollars and 2 shillings, or 33| cents. 13. If 7 pounds cost 40 cents, 1 will cost 5f cents; 10 pounds will cost 57f cents. 16. 1 cock would empty it in 6 hours, and 7 cocks Sect. 11.] KEY. 161 would empty it in } of 6 hours, or ^ of 1 hour, which is $ of 60 minutes; f oMBO minutes; is 51f minutes. SECTION XI. A. 2. 2 halves of a number make the number; consequently 1 and 1 half is the half of 2 times 1 and 1 half, which is 3. 15. 4^ is £ of 5 times 4 and 3, which is 22|. 17. 4$ is i of 9 times 4$, which is 39f B. 4. 5 is 3 times | of 5, which is f, or If. 30. If 8 is £ of some number, | of 8 is { of the same number. | of 8 is 2|, 2f is { of 4 times 2|, which is 10}; therefore 8 is £ of 10}. 40. If 8 is I of 8 is |; } of 8 is f, | is | of y, or Of; therefore 8 is ^ of 9 f. 52. If £ of a ton cost 23 dollars, £ of a ton must be ± of 23, that is 4| dollars, and the whole would cost 9 times as much, that is, 4 If. 69. i of 65 is 7f; 7f is \ of 5 times 7|, which 36|. 65isfof36£. C. 4. 37 is f of 32f, which taken from 37 leaves 4£. Ans. 4£ dollars. 5. 7 feet must be | of the whole pole. f 6. If he lost }, he must have sold it for £ of what it cost. 47 is $ of 60f Ans. 60 dollars and 42f cents. Miscellaneous Examples. £ 1. The shadow of the staff is f of the length of the staff; therefore the shadow of the pole is J of the length of the pole. 67 is £ of 83£. Ans. 83} feet. 2. 9 gallons remain in the cistern in 1 hour, will be filled in 10 hours and |; £ of 60 minutes 14* 162 KEY. [Part*. are 46 minutes anii £; £ of 60 seconds are 40 sec- onds. Arts. 10 hoirs, 46 minutes, 40 seconds. 10. Find f of 33, and subtract it from 17. Arts. 3$. 11. It will take 3 times 10 yards. 13. 5 is f of 3; it will take f as much. Or 7 yards, 5 quarters wide, are equal to 35 yards 1 quar- ter wide, which is equal to llf yards that is 3 Quarters wiHp quarters wide^ 15. | of 37 dollars 16. f as much. SECTION XII. The examples in this section are performed in precisely the same manner, as those in the sections to which they refer. All the difficulty consists in i comprehending, that fractions expressed in figures ^signify the same thing as when expressed in words. KMake the pupil express them in words, and all the rlifficulty will vanish. Let particular attention be paid to the explanation of fractions given in the section. VIII. A. 6. In 7 how many £ 1 expressed in words, is, in 7 how many sixths? Ans. %2. 14. Reduce 8f37 to an improper fraction; that is, in 8 and 3 tenths, how many tenths? Ans. f §v B. 8. 2T3 are how many times 11 That is, in 23 sevenths how many whole ones 1 Ans. 3f. , EXy^B. 3. How much is 5 times 6^? That is, how much is 5 times 6 and 4 sevenths 1 Ans. 32f. V. & X. 15. What is f of 27? That is, what is 5 eighths of 27 1 Ans. 16f | VI. & XI. A. 8. 7j is | of what number? That & is, 7 and 6 sevenths is 1 eighth of what number 1 I Ans* 62f Sect 12.] 163 KEY. 6. 4. 12 is f of what numb^f That is, 12 is 3 sevenths of what number 1 AnWm&* 12. 4 is f of what number? That is, 4 is 3 fifths of what number ? Ans. 6|. Explanation of Plate HZ Plate III is intended to represent fractions of unity, divided into other fractions ; it is, therefore, an extension of plate II. It differs from it, only in this, that besides the vertical divisions, the squares are divided horizontally, so as to cut the fractions of the square into fractions of fractions. The horizontal lines are dotted, but they are to be considered as lines. This plate, like the preceding, is divided into ten rows of squares, each row containing ten equal squares. In the first row, the first square is undi- vided, the 9 following squares are divided by hori- zontal lines into from two to ten equal parts. In all the other squares the vertical divisions are the same as in Plate II, and besides this, each row is divided horizontally in the same manner as the first row. By means of this double division, the 2d row pre-* sents a series of fractions, from halves to twentieths. The 3d row presents a series from thirds to thirti- eths, and so on to the 10th row, which presents a series from tenths to hundredths. The 2d row, besides presenting halves, fourths, sixths, eighths, and £ is 5 times as much as £, therefore \ of £ is f^. This may be readily seen on the plate. In the sixth row, third square, find £ by the vertical division, then these being divided each into three parts by the hori- zontal division, and \ of each bein^ taken, you will have T5T. 52. In the 4th row, the 3d square shows that \ of \ is T*y, and | must be twice as much, or -fa. 56. In the fifth row, the 3d square shows that \ of § is XV but f must be twice as much as £9 there- fore | of f, are 168 mKEY [PaM% 78. 8f isy,)oryu h 79. 8f is V, | of | is TV, consequently j of y is f£, or Ijf 86. We may say | of 8f is 2, and 2£ over, then 2$ is V, and £ of y is §f, hence % of 8£ is 2ff. 90. \ of 18| is 2|J, and ^ is 3 times as much, or m B. 4. It would take 1 man 4 times 9f, or 374 days, and 7 men would do it in \ of that time, that is, in 5Jf days. SECTION XV. A. This section contains the divisions of whole numbers by fractions, aaid fractions by fractions. 1. Since there are f in 2, it is evident that he could give them to 6 boys if he gave them £ apiece, but if he g$ve them § apiece, he could give them to only one half as many, or 3 boys. 5. If | of a barrel would last them one month, it is evident that 4 barrels would last 20 months, but since it takes f of a barrel, it will last them but one half as long, or 10 months. 7. 6f is y. If | of a bushel would last a week, 6f bushels would last 27 weeks; but since it takes £, it will last only £ of the time, or 9 weeks. 13. If he had given £ of a bushel apiece, he might Have givej^it to 17 persons, but. since he gave 3 halves apiece, he could give it to only £ of that number, that is to 5 persons, and he would have 1 bushel left, which would be § of enough for another. 23. 9f is \6, and 14 is y. If it had been only I of a dollar a barrel, he might have bought 66 barrels for 9f dollars, but since it was y a bar- SkcL 15.] KBIT 169 rel, he could buy only yV of that number, that is, 6 barrels. 25 and 26. Ans. Of. 31. 4J is Vt *t is ¥• Now £ is contained in y 48 times, and %l is contained only ^T part as many times, consequently only 2/T or 2$. B. 1. £ is ^; consequently 5 pounds can be ^ bought for | of a dollar. 3. J is T9s, and £ is -fa. If he had given only T*7 apiece, he could have given it to 9 persons, but since he gave y% he could give it to only 1 half as many, or 4£ persons. 5. ^ is /r, and f is £f. If a pound had cost JT of a dollar, 14 pounds could be bought for |f of a dollar, but since it costs ¥3T, only £ as many can be bought; that is, 4| pounds. 9» | is ih and H is H* a bushel had cost TV of a dollar, 65 bushels might have been bought, but since it cost only Ty part as much could be bought; that is, 4T'F bushels. 12. % is T\, and £ is jf, T'T is contained in |f 15 times, but is contained only £ as many times; that is, 3f times. Miscellaneous Examples, 5. £ of ,a penny is f of 4 farthings. jins. 2} farthings. 6. | of 12 pence. Arts. 10 pence. 7. | of 4 quarters is 2 quarters and f of a quar- ter ; f of a quarter is f of 4 nails, which is If nails. Ans. 2 quarters, lg nails. 13. f of 24 hours is 15 hours. 14. | of 24 hours is 14 hours and § of an hour; £ of 60 minutes is 24 minutes. Ans. 14 hours, 24 minutes. 15 170 [Part 2. KEY. 28. There being 4 farthings in a penny, 1 far- thing is \ part of a penny. ^ 30. 3 farthings is f of a penny. 31. 1 penny is y1^ of a shilling, because there are 12 pence in a shilling. 34. 5 pence is T5¥ of a shilling. 41. 1 shilling is ^ of a pound. 43. 3 shillings is ^ of a pound. 48. 1 farthing is of one shilling. 49. 2 farthings is TaF, or ^ of a shillkig. 5 far- things is of a shilling. 51. 1 penny is of 1 pound. 7 pence is of l£. 59. 3s. 5d. is 41 pence, which is -fife of 1£. 75. 1 nail is T!g of a yard, 5 nails is T5F of a yard. 89. 1 oz. is TV of 1 lb. 15 oz. is }£ of 1 lb. 91. 1 lb. is of 1 quarter. 9 lbs. is & of 1 quarter. 100. £t the end of 1 hour they would be 7 and f miles apart. In 7 hours, 7 times 7J, which is 54f miles. 121. This is the principle of fellowship; 3 shil- lings were paid; one paid the other f. 122. One paid f, the other f. 123. 20 dollars were paid in the whole, one paid another ^ and the third 121. 3 and 4 and 5, are 12. The first put in T\; the second T4*; the third T5^. 129. 4 dollars for 2 months, is the same as 8 dol- lars for 1 month; 3 dollars for 3 months, is the same as 9 dollars for 1 month; and 2 dollars for 4 months, is the same as 8 dollars for 1 month. The question is the same as if A had put in 8 dollars, B 9 dollars, and C 8 dollars. A must have ¥8J9 B A, and C ¥8T, of 100 doUars. 131. A's money was in 4 times as long as C's. It is the same as if A had put in 8 dollars for the Part 2.] 171 KEY. same time, and B 8 dollars for the same time. A must have Jfe, B ^, and C of 88 dollar^ The examples 127, 128, 129, 130, and 491, are double or compound fellowship. 139. The interest of 50 dollars for I year and 6 months is 4 dollars and 50 cents, and for 1 month it is 25 cents. The interest of 7 dollars for 18 months (a dollar is \ of a cent a month) is 63 cents. The whole amounts to 5 dollars and 38 cents. 140. The interest of 200 dollars for \\ years is 16 dollars. The interest of 67 dollars is 67 cents for every 2 months, for 16 months it will be 8 times 67 cents, which are 5 dollars and 36 cents. The whole interest is 21 dollars and 36 cents. 143. The interest of 100 dollars for 2{ years, is 13 dollars and 50 cents. The interest of 100 dollars for 60 days would be 1 dollar, the interest for 20 days will be } of a dollar, or 33£ cents. The interest of 1 dollar for 2| years is 13j cents, for 10 dollars the interest would be 1 dollar and 35 cents, and for 30 dollars, 4 dollars and 5 cents. The interest of 7 dollars for 2} years is 7 times 13£ cents or 94^ cents. The interest of 37 dollars for 60 days would be 37 cents, and for 20 days £ of 37 cents, or 12£ cents. The whole interest is 18 dollars and 95£ cents. 146. They would both together do f of the work in 1 day, and it would take them | of a day to do the other Ans. 1£ day. 147. f would be done in 1 day, and it would take £ of a day to do the other Ans. 1£ days. 149. They both together consume \ of a bushel in a week, but the woman alone consumes only \ of a bushel in a week. That is, they both together consume f ^ in a week, but the woman alone only -jSg-, consequently, the man alone would consume fV» and a bushel would last him 3£ weeks. 172 KEY. [Part% and B can build \ of it in 1 day; A, B, and O^fean build \ of it in 1 day, the difference be- tweetffNind \ is T\ ; therefore C can build T37 of it in 1 day; and it would take him 13£ days to build it alone. 164. Find how much they might eat in a day, in order to make it last 1 month, and then it will be easy to find how much they may eat in a day to make it last 11 months. 167. The money is 7 parts of the whole, and the purse one part; consequently the money is J, and the purse \ of 16. 170. He gave 1 part for the apple, 2 parts for the orange, and 4 parts for the melon. These make 7 parts. The apple 3 cents, the orange 6 cents, and the melon 12 cents. 175. If to a number half of itself be added, the sum is f of that number; hence, subtract 2j from 100 and the remainder is f of the number of geese that he had. 180. This must be reduced to 6ths. 1 half is f, and \ is f, and the number itself is f. If therefore to the whole number its half and its third be added, the sum will be y ; hence, 77 is y 0I" the number. 181. \ is |; therefore if to a number \ and \ of itself be added, the whole number will be \; but when 18 more is added to }, the first number is doubled; that is, the number is } of the first num- ber; therefore 18 is \ of the number