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 4, 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 3, 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. 75. It is important to observe, that any division, whether it can be performed in whole numbers or not, may be indicated by a fractional expression; 3, for instance, expresses evidently the quotient of 36 by 3, as well as 12, for being contained three times in unity, 36 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. Arith. 7 Also, the difference between 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 denominator, 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. 17, 38. 36 to a common denominator. Ans. 48, 48. 30 Reduce and Reduce, 3, 4, and to a common denominator. 0 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 24ths 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,, 11, 11. 0 49 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 are to be added to them, or subtracted from them; and if the whole numbers are accompanied with fractions, they must be reduced to the same denominator with these fractions. It is thus, that the addition of four units and changes itself into the addition of 2 and, and gives for the result. To add 3 to 54, the whole numbers must be reduced to fractions, of the same denomination as those which accompany them, which reduction gives 3 and 4; with these results the sum is found to be 55, or 84. If, lastly, were to be subtracted from 31, the operation would be reduced to taking from 3, and the remainder would be . 82. The rule given for the reduction of fractions to a common denominator supposes, that a product resulting from the successive multiplications of several numbers into each other, does not vary, in whatever order these multiplications may be performed; this truth, though almost always considered as self-evident, needs to be proved. We shall begin with showing, that to multiply one number by the product of two others, is the same thing as to multiply it at first by one of them, and then to multiply that product by the other. For instance, instead of multiplying 3 by 35, the product of 7 and 5, it will be the same thing if we multiply 3 by 5, and then that product by 7. The proposition will be evident, if, |