LESSONS IN ARITHMETIC.-No. XXI. (Continued from p. 232.) PROBLEMS IN VULGAR FRACTIONS. 14. In our last lesson, we intentionally omitted the answers to three of the exercises, in order to bring out the skill of our students. We now give the solutions of them for comparison with their own. In Ex. 12, the greatest common measure of the terms of the fraction is 1, therefore the fraction is already in its lowest terms. In Ex. 13, the fraction; so that the individual holds threeeighths of the whole concern. In Ex. 14, the fraction 868; so that three-fifths of the ship belongs to the contributor. PROBLEM II. To reduce given fractions having different denominators into fractions of equal value having a common denominator. DEFINITION 6.-When fractions have all the same denominator, they are said to have a common denominator; the meaning is, that since the denominator belongs to them all, it is therefore common to them all. Thus, the fractions,,,, have all the same denominator 8; therefore these fractions are said to have a common denominator, that is, they all express various parts or portions of a unit, but these, though different in number, are all of the same kind. The first expresses three-eighths of a unit, the second expresses five-eighths of a unit, &c.; but they all contain parts of the same kind, that is, a certain number of eighths. RULE 1.-Multiply both terms of each fraction by the product of the denominators of all the other fractions, and the results will be the fractions required. The reason of this rule is evident from the principle (p. 134, art. 10) that a fraction is neither increased nor diminished in value by multiplying both of its terms by the same number; and also, from the principle (p. 95, col. 2, prin. 2) that the continued product of the factors of any number will be the same in whatever order they may be taken. EXAMPLE. Reduce the fractions,, and to fractions of equal value, but having a common denominator. EXPLANATION. Here, the operation of multiplying each fraction by the product of the denominators of all the other fractions (or, rather, by these denominators themselves), is exhibited in the middle column by indication of the factors; and the necessity of the common denominator 24, being the result of the same factors in each case is rendered more manifest. When the given fractions are numerous or large, the process may be shortened by the following method: together for the common denominator. Then, divide this common RULE 2.-Multiply all the denominators of the given fractions denominator by each of the denominators, and the quotients will be the multipliers for both terms of the fractions to which these denominators belong, which being multiplied thereby will give the fractions required. | The reason of this rule is evident from the fact that the common multiple of any set of numbers is divisible (without a remainder) by each of them (p. 57, col. 1); and that the quotient multiplied by the divisor reproduces the dividend; not forgetting the principle that when both terms of a fraction are multiplied by the same quantity its value is not altered. Here, the product of all the denominators of the given fractions is 60, which is the common denominator. Next, the quotients arising from the division of this common denominator by each of the denominators in succession, are 20, 15, and 12. Lastly, these quotients being employed as multipliers to both terms of the frac In this example, both terms of the first given fraction are multi-tions whose denominators respectively produced them, give the midplied by 12, which is the product of the denominators 3 and 4 of die column of the above operation by indication of the factors, and all the other given fractions, and the result is, the first fraction the third and last column as the results, which are the fractions required. Next, both terms of the second given fraction are multi-required. Hence, the answer to the question is 8, 8, and 13. plied by 8, which is the product of the denominators 2 and 4, of all the other given fractions, and the result is, the second fraction required. Again, both terms of the third given fraction is multiplied by 6, which is the product of the denominators of all the other given fractions, and the result is 1, which is the third fraction required. Hence, the answer is 4, 14, and 14, for these fractions have the common denominator 24. And the reason of this is plain, from the second principle above mentioned, and from the manner in which they were produced: for 1st 16 Denominator 24=12 × 2=3×4×2 24 8X32×4X3 24 6X4=2X3 X4 2nd 3rd NOTE.-When fractions are given to be reduced to fractions of equal value having a common denominator, it is necessary and proper, in order to save labour in the operation, always to reduce any of them that will admit of it, to their lowest terms before you apply either of the preceding rules. For instance: if the fractions 18, 193, and 5000 were given to be reduced to a common denominator, and you were at once to apply either of the preceding rules to the reduction of these fractions, the operation would be needlessly laborious, and the results would assume the following cumbrous form:-117888888, 287296000, 321861000, and 134288888 21824999. Now, if the given fractions had been reduced to their lowest terms by Problem 1., they would have assumed the forms of, 3, and 4; then, by either of the preceding rules, when reduced to a common denominator, they become 4, 4, and, for the fractions required. It frequently happens, however, that fractions are given to be reduced to a common denominator, whose denominators have some common factors, and in such cases the operation of reducing to a common denominator can be greatly abridged. The method of performing this abridgement will be best understood by proceeding according to the following rule :→ RULE 3.-Find the least common multiple of all the denominators of the given fractions. Then, divide this least common multiple (which is called the least common denominator) by cach of the denominators, and the quotients will be the multipliers for both terms of the fractions to which the denominators belong, which being multiplied thereby will give the fractions required. EXAMPLE. Reduce the fractions 1, 4, 4, 4, 3, and to fractions of equal value having a common denominator. Here, the denominators are 2, 4, 5, 6, 8, and 10; rejecting 2, 4, and 5, as submultiples of 4, 8, and 10, we have to find the least common multiple of 6, 8, and 10 (p. 57, col. 1). This, by the rule formerly given, is 120, as follows: 2) 6, 8, 10 3, 4, 5 2 X3 X4 X5 = 120, least common Observe here, that as there is no common measure to any two of Now, dividing the least common denominator, by each of the denominators of the given fractions, we have the quotients as follows:- 120 120 260 4 30 120 120 524 quotients and 120 8=15 12010-12} And multiplying both terms of each of the fractions to which these Here, by reducing the two fractions to a common denominator by Rule 1 or 2, we have 142 9 10 72 80 142 But from this operation, it plain that is greater than 8; there fore, the fraction is greater than. Moreover, the difference between and 8, is; therefore is greater than by the fraction. Next, to find their sum, we have +18= 1188; whence the sum of the two fractions is or 18, which by reduction is the same as 14. In like manner you may find the greatest or the least of any number of given fractions; for, by reducing them to a common denominator, you will see at once that the fraction whose numerator is then the greatest, is the greatest fraction; and the fraction whose numerator is the least, is the least fraction. You can also find the sum of any number of fractions, and the difference between any two of the same. EXERCISES. quotients belong, we have the fractions required, as in the following value, and having a common denominator : Reduce the following given fractions to fractions of the same operation : A moins que vous ne preniez bien rice. Quel indigne plaisir peut avoir l'ava- He will do nothing of the kind, unless offer? What is the use of hoarding up unless we enjoy. The principal use of this problem in Vulgar Fractions, is to prepare any set of given fractions for addition or subtraction; and the reason why this rule is necessary is simply this: that it is only EXAMPLES ILLUSTRATING THE VARIOUS USES OF THE PRINCIPAL like things which can be added or subtracted. Now, unless that fractional parts or denominations of any sort are precisely of the same kind, they cannot be added together or subtracted; for if this were done the answer would have no meaning. Thus, for instance, we could not add to, or subtract from, because the denominators of these fractions are different; neither could we tell by any numerical comparison of which they are at present capable, which of the two is the greater. But if these two fractions, and, be brought or reduced to fractions of equal value, having a common denominator, then we can immediately institute a comparison between them, and tell not only which of the two fractions is the greater, but also what is the sum of and the difference between the two fractions. Thus, by reducing and to a common denominator by any of the preceding rules, we find that they become and. Now here it is manifest that is greater than &; therefore is greater than and if the question be asked how much is greater than, the answer is plainly because -=—=; for taking two-sixths from three-sixths, leaves one-sixth. In like manner, if the sum of the two fractions be required, we have +=; that is, that threesixths and two-sixths added together make five-sixths. In this case, as in every other of a similar kind, it is plain that the sum or difference between the two fractions when reduced to a common denominator by this problem, is found at once by adding together the numerators, or by subtracting the one numerator from the other; and then putting the common denominator under the sum or the difference thus obtained, for the answer. Si ce n'est toi, c'est donc ton frère. DE CUSSY. you should love him. I am, therefore, a witness of their If it is not you, then it your brother. I have none. Then it must be some DE MEME QUE. ET. NI. with Me! Tis alcep is disturbed redher by the jean, nor by the cheful desires of quote. Heureux colu! qui sait se contenter | Happy is he who can content himself On n'est jamais si heureux, ni si LA ROCHEFOucauld. MASSILLON. Or sus, mettons nous à l'ouvrage. La fortune, soit bonne ou mauvaise, La liberté de publier ses penscér, OR. ov. You lose thus the confidence of your friends without having rendered them either better or more skilful. happy as we fancy. We are never so happy nor so un That holy law Inows no longer either poor or rich, noble or plebeian, master or slave. Fortune, be it good or bul, be it transient or constant, has no power over the soul of the wise. PARCEQUE. As the sun shines upon earth, so will Les grands hommes entreprennent | Great men undertake great things the just shine in heaven. de grandes choses parcequ'elles To be inaccessible and proud is to be Là tout est beau, parceque tout est Une famille vertueuse est un vais-4 seau tenu pendant la tempête par deux ancres, la religion et les CHATEAUBRIAND. mœurs. Quel carnage de toutes parts ! vieillards, Et la sœur et le frère, Et la fille et la mère, Le fils dans les bras de son père. Lorsque l'innocence terre. vrai. because they are great; and fools because they believe them easy. cause everything is true, Every thing there is beautiful, be J. J. ROUSSEAU. POURTANT. less, its elevation. strengthened during the tempest by Le style le moins noble a pourtant | The least elevated style has, neverthe two anchors, religion and morals. What carnage on all sides! LORSQUE-QUAND. habitait la When innocence inhabited the earth. Quand vous me haïriez, je ne m'en RACINE. plaindrais pas. Quand nous n'aurions égard qu'au repos seul de notre vie, quand nous n'aurions point d'autre intérêt icibas que de nous préparer des jours heureux, quel bonheur de prevenir d'avance et d'étouffer dans leur naissance tant de passions violentes. MASSILLON. If even you hated me I would not If even we considered merely the re- MAIS. which we have since seen him possessed, but with the consent of the people, with forms less regal, but perhaps more worthy. but the mind, C'était dejâ la puissance impériale | It was already the imperial power of Jamais on ne vit un si grand ex- Nous n'avons que peu de temps à . fier? Qu'il fasse ce qu'il lui plaira. To keep on the defensive is a wise There is no but" in the matter; GANGANELLI. Never was such a striking example seen, that courage is not incompatible with effeminacy. fying myself Let him do what he pleases. ful evenings! Do disease of the mind. than the Gospel. There will never be any better guide Vous le voulez ? ainsi soit il! Un mal funeste et contagieux se répandit dans les principales villes de la Normandie; soit que l'intempérie des saisons eût laissé dans les airs quelque maligne impression, soit qu'un commerce fatal eût apporté des pays éloignés, avec de fragiles richesses, des semences de maladie et de mort, soit que l'ange de Dieu eût éntendu la main pour frapper cette malheureuse province. FLECHIER. Whether he does it, whether he does O. 8. Ouest-sud. it not. P. S. Post-scriptum, Be it the boldness of the enterprise, S. A. S. Son Altesse Sérénissime. Be it for good, be it for evil, my S. Em. Eminence. You will have it so? So be it? S. Ex. Son Excellence. A fatal and contagious disease spreads. M. Sa Majeste. S. M. B. Sa Majesté Britannique. S. M. T. F.Sa Majesté Très Fidèle. S. O. Sud-ouest. S. P. Saint Père. SS. PP. Les Saints Pères. S. S. Sa Sainteté. S. S. E. Sud-sud-est. S. S. O. Sud-sud-ouest. West. [We have now arrived at the end of the FIRST PART of our course of LESSONS IN FRENCH, and next week we shall begin the SECOND PART. As this Second Part was constantly referred to in the First Part, by sectional references, thus [§ 13, (2.)], see page 31, vol. I., col. 1, line 11; and as it contains a systematic treatise on FRENCH GRAMMAR, with all the usual rules of Etymology and Syntax, it will be not only a necessary sequel to that part, but an essential desideratum in order to render our students complete French scholars. They will do well, therefore, to revise all their past lessons, and re-examine all their former difficulties by the new light which will thus be thrown upon the study of the French Language, and particularly upon its very peculiar idioms, &c.] CORRESPONDENCE MANLY SPIRIT. SIR,--Permit me to offer you my sincere and hearty thanks for the benefit I have derived from the POPULAR EDUCATOR, and to assure you that I am extremely happy in availing myself of the advantages of your exertions in the cause of learning. I study with great ardour, and am possessed with so keen a relish of learning, that animum in literis perpetuo intendo. Being only a journeyman shoemaker, some of my acquaintances consider that "college larnin" is above me. Truly so, and therefore I will climb up to it. This I find, that be a man's position in society as humble as it may, learning makes him feel himself a gentleman, and confers upon him genuine respectability. The inclosed poem may interest some of your students of Latin. It was written by Sir Thomas More, Lord High Chancellor of England, in 1530, and addressed to a friend of his who was in search of a wife. The regularity of the measure will partially assist the right pronunciation of the words. Each student can make a translation for himself; and some one will perhaps send you a poetical translation. DOMINIC DIVERS. Christmas day, 1852. Yours sincerely, DE UXORE. "Proculque stulta sit, Parvis libellulis ; Semper loquacitas ; Semper, nec unquam erit Dum plectra personat; Jam cum omnium gravi Talem olim ego putem Nec unquam ab inferis Nec profuit minus CONSTRUCTION OF THE VIOLIN. SIR,-In answer to John Stocking's inquiry, "What is the best deal for the belly of a violin?" I would say, for the belly, good Swiss pine cleft from the heart to the bark. The wood should be cut in the month of December or of January, and only the south side of the tree should be used. It must be split in such a manner as to leave each piece an inch thick at the bark-side, and a quarter of an inch at the heart of the tree. The back of the best Cremonese are as often made in one piece as in two. When jointed, the wood cuts to more advantage. But John Stocking should buy "Otto on the Structure of the Violin" (Cocks and Co., price 3s.), where he will find a diagram for striking a fine model for violins, and all the information that he wants. I am only an amateur, but am well acquainted with the violins of the old makers. I beg to remain, your constant reader, H. B. BUSH. Long Wittenham, Abingdon, Berks., Jan. 2, 1853. CO-INSTRUCTION SOCIETIES. A "Young Man's Commercial and General Improvement Society" has been formed in Dublin, having for its objects-(1), Education (by lectures, formation of classes, and conversation) in the principles and practice of trade and commerce; (2), the study of special branches of knowledge likely to be useful to young commercial men; and (3), the mutual improvement of all by means of friendly meetings and intellectual intercourse. The main object of the society is avowed to be-the association of young men connected with the mercantile and commercial offices and establishments in Dublin; but no young man d eirous of advancement in life will be excluded from the society. The greater variety of avocations represented in such a society, the more it will resemble the business world, and, therefore, the more useful will it be. Several literary gentlemen of eminence in Dublin have rendered advice and assistance; particularly the editor of the Commercial Journal. 13, Anglesea-street, Dublin. LITERARY NOTICES. Part I. THE ILLUSTRATED EXHIBITOR AND MAGAZINE OF ART-A new and improved Series of this work, under the title of the ILLUSTRATED MAGAZINE OF ART, has just commenced. Each Number is now enclosed in a neat wrapper, price 3d. In addition to numerous Engravings in the text, each number contains a fine Engraving, worked on Plate Paper. With Number 1, was presented, gratis, a splendid View of the Interior of St. Paul's Cathedral, during the Interment of the late Duke of Wellington, printed upon fine Plate Paper, measuring eighteen inches by thirteen. This Engraving alone is worth four times the cost of the Number of the MAGAZINE OF ART.-Part I., price ls., will be ready February 1st. The large Engraving of St. Paul's will also be presented with every copy of this Part. THE ALTAR OF THE HOUSEHOLD; or, DOMESTIC WORSHIP. s now ready, price 1s. 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