71. It may sometimes happen that two mixed numbers, or whole numbers joined with fractions, are to be multiplied, one by the other, as, for instance, 3 by 43. The most simple mode of obtaining the product is, to reduce the whole numbers to fractions by the process in article 69; the two factors will then be expressed by 2 and 4, and their product, by 14 or 1819, by extracting the whole ones (68). 1144 63 72. The name fractions of fractions is sometimes given to the product of several fractions; in this sense we say, of . This expression denotes of the quantity represented by of the original unit, and taken in its stead for unity. These two fractions are reduced to one by multiplication (70), and the result, 1, expresses the value of the quantity required, with relation. to the original unit; that is, of the quantity represented by of unity is equivalent to of unity. If it were required to take 8 of this result, it would amount to taking of of, and these fractions, reduced to one, would give for the value of the quantity sought, with relation to the original unit. 73. The word contain, in its strict sense, is not more proper in the different cases presented by division, than the word repeat in those presented by multiplication; for it cannot be said that the dividend contains the divisor, when it is less than the latter; the expression is generally used, but only by analogy and extension. To generalize division, the dividend must be considered as having the same relation to the quotient, that the divisor has to unity, because the divisor and quotient are the two factors of the dividend (36). This consideration is conformable to every case that division can present. When, for instance, the divisor is 5, the dividend is equal to 5 times the quotient, and, consequently, this last is the fifth part of the dividend. If the divisor be a fraction, for instance, the dividend cannot be but half of the quotient, or the latter must be the double of the former. to be only The definition, just given, easily suggests the mode of proceeding, when the divisor is a fraction. Let us take, for example, . In this case the dividend ought of the quotient; but being of, we shall have of the quotient, by taking of the dividend, or dividing it by 4. Thus knowing of the quotient, we have only to take it 5 times, or multiply it by 5, to obtain the quotient. In this operation the dividend is divided by 4 and multiplied by 5, which is the same as taking of the dividend, or multiplying it by, which fraction is no other than the divisor inverted. This example shows, that, in general, to divide any number by a fraction, it must be multiplied by the fraction inverted. For instance, let it be required to divide 9 by ; this will be done by multiplying it by, and the quotient will be found to be 36 or 12. Also 13 divided by will be the same as 13 multiplied by or . The required quotient will be 18, by extracting the whole ones (68). It is evident that, whenever the numerator of the divisor is less than the denominator, the quotient will exceed the dividend, because the divisor in that case, being less than unity, must be contained in the dividend a greater number of times, than unity is, which, taken for a divisor, always gives a quotient exactly the same as the dividend. 74. When the dividend is a fraction, the operation amounts to multiplying the dividend by the divisor inverted (70). Let it be required to divide by ; according to the preceding article, must be multiplied by 2, which gives . It is evident, that the above operation may be enunciated thus ; To divide one fraction by another, the numerator of the first must be multiplied by the denominator of the second, and the denominator of the first, by the numerator of the second. If there be whole numbers joined to the given fractions, they must be reduced to fractions, and the above rule applied to the results. whether it 75. It is important to observe, that any division, can be performed in whole numbers or not, may be indicated by a fractional expression;, for instance, expresses evidently the quotient of 36 by 3, as well as 12, for being contained three times in unity, will be contained 3 times in 36 units, as the quotient of 36 by 3 must be. 76. It may seem preposterous to treat of the multiplication and division of fractions before having said any thing of the manner of adding and subtracting them; but this order has been followed, because multiplication and division follow as the immediate consequences of the remark given in the table of article 55, but addition and subtraction require some previous preparation. It is, besides, by no means surprising, that it should be more easy to multiply and divide fractions, than to add and subtract them, since they are derived from division, which is so nearly related to multiplication. There will be many opportunities, in what follows, of becoming convinced of this truth; that operations to be performed on quantities are so much the more easy, as they approach nearer to the origin of these quantities. We will now proceed to the addition and subtraction of fractions. 77. When the fractions on which these operations are to be performed have the same denominator, as they contain none but parts of the same denomination, and consequently of the same magnitude or value, they can be added or subtracted in the same manner as whole numbers, care being taken to mark, in the result, the denomination of the parts, of which it is composed. It is indeed very plain, that and make, as 2 quantities and 3 quantities of the same kind make 5 of that kind, whatever it may be. Also, the difference between 3 and is, as the difference between 3 quantities and 8 quantities, of the same kind, is 5 of that kind, whatever it may be. Hence it must be concluded, that, to add or subtract fractions, having the same denominator, the sum or difference of their numerators must be taken, aud the common denomtor written under the result. 78. When the given fractions have different denominators, it is impossible to add together, or subtract, one from the other, the parts of which they are composed, because these parts are of different magnitudes; but to obviate this difficulty, the fractions are made to undergo a change, which brings them to parts of the same magnitude, by giving them a common denominator. For instance, let the fractions be and; if each term of the first be multiplied by 5, the denominator of the second, the first will be changed into ; and if each term of the second be multiplied by 3, the denominator of the first, the second will be changed into ; thus two new expressions will be formed, having the same value as the given fractions (56). This operation, necessary for comparing the respective magnitudes of two fractions, consists simply in finding, to express them, parts of an unit sufficiently small to be contained exactly in each of those which form the given fractions. It is plain, in the above example, that the fifteenth part of an unit will exactly measure and of this unit, because contains five 15ths, and contains three 15ths. The process, applied to the fractions and, will admit of being applied to any others. In general, to reduce any two fractions to the same denominator, the two terms of each of them must be multiplied by the denominator of the other. 79. Any number of fractions are reduced to a common denominafor, by multiplying the two terms of each by the product of the denominators of all the others; for it is plain that the new denominators are all the same, since each one is the product of all the original denominators, and that the new fractions have the same value as the former ones, since nothing has been done except multiplying each term of these by the same number (56). Examples. Reduce and to a common denominator. Ans. 37, 38. 20 361 36 78, 48. 48 Reduce and to a common denominator. Ans. $8. Ans. 630 1 8 9 0 1800 1750 3150 31509 31509 3150 The preceding rule conducts us, in all cases, to the proposed end; but when the denominators of the fractions in question are not prime to each other, there is a common denominator more simple than that which is thus obtained, and which may be shown to result from considerations analogous to those given in the preceding articles. If, for instance, the fractions were,, ,, as nothing more is required, for reducing them to a common denominator, than to divide unity into parts, which shall be exactly contained in those of which these fractions consist, it will be sufficient to find the smallest number, which can be exactly divided by each of their denominators, 3, 4, 6, 8; and this will be discovered by trying to divide the multiples of 3 by 4, 6, 8; which does not succeed until we come to 24, when we have only to change the given fractions into 24th of an unit. To perform this operation we must ascertain successively how many times the denominators, 3, 4, 6, and 8, are contained in 24, and the quotients will be the numbers, by which each term of the respective fractions must be multiplied, to be reduced to the common denominator, 24. It will thus be found, that each term of must be multiplied by 8, each term of by 6, each term of by 4, and each term of by 3; the fractions will then become, H. 11. 18 20 3}• Algebra will furnish the means of facilitating the application of this process. 80. By reducing fractions to the same denominator, they may be added and subtracted as in article 77. 81. When there are at the same time both whole numbers and fractions, the whole numbers, if they stand alone, must be converted into fractions of the same denomination as those which |