MATHEMATICS. ALGEBRA. 



[MUCTIOXS ADDITION. 



(r o) z* a', tho Mcood fraction will be converted 

 into ono with a denominator the aamo M the first, l>\ 

 wanly multiplying numerator and denominator of tlio 



wood by x-o, .-. the fraction* are 



a 1 * -- i 



denominator is 6, the least number divisible by 

 . ata 3, 2, and C; the letters are oV, the least 

 quantity diruiblo by a*6>, a%*, and 06, .'. tho changed 



fractions are ^ + ^ + ^f; each fraction here 



differing from that which it replaces only in appearance 

 it is tho original fraction with numerator and denomi- 

 nator multiplied by the same thing. 



It will not bo necessary that wo should give any 

 exorcises expressly for practice in tho preceding rule. 

 The examples furnished in addition and subtraction 

 the rule for which we are now about to give, will equally 

 afford practice in the foregoing operation ; for, us you 

 have already aeon, all fractions must have a common 

 denominator before they can be either added or sub- 

 tracted. We shall therefore at once proceed to 



ADDITION AND 8UBTBACTION OF FRACTIONS. 



RfLE. Addition. Reduce the fractions to equivalent 

 trnos with a common denominator, which place under 

 the mm of tho changed numerators. 



Subtraction. Reduce the fractions to equivalent 

 ones with a common denominator, which place under 

 the difference of tho changed numerators. 



1 * + " A. b _i_ 1 15(s-r 



2 r 3 6 ;;o 



x) 



3x 



o 6 



3* 



a* 6 



a 3 6 a o s 6" 



3 (4s + 2)j ~ _^ 

 Sx 



:i.r 



a 





In this example, as well as in tho preceding, an integer 

 occurs iu connection with a fraction ; and it may here 

 !> noticed, tliat an integer may always bo put in tho 

 form of a fraction by simply giving to it 1 for denomi- 

 nator: thus, in tho present example, the 2 may be 



regarded as ?. 



EXAMPLES FOB EXERCISE. 



Van. It sometimes happens that the mm or difference of fraction! b a 

 friction rath tkmt mncntor and denominator hare a factor common to 

 both. Such common (acton should always be expunged from the final 

 nralt, whentrtT thjr an seen to . -ivetothe 



Uoo an appearance unneeea*rr complicated. A fraction thus 

 o|irld of all common facton In nnmrrator and denominator U aaid to 



he In iu latent trrmi, the numerator anil denominator bi inn called the 

 ttrmt of tki fraction. 



I 





1 | i- 



1 



13. Shaw that 



NOT*. The student may obtain considerable aamistaaco In tho study of 

 tlii- addition, &c., of aliebraic fractions, by combining with It that of I in 

 arithmetical branch of the same subject.* ED. 



MULTIPLICATION OF FRACTIONS. 



If a fraction - - is to be multiplied by an integer c, we 

 have to take a things, of denomination b, c times ; the pro- 

 duct is therefore -.- ; that is, ac of those tilings, just as 



if we had to multiply a pounds, or o ounces, by c, tho 

 product would be ac of those pounds or ounces. But, if 

 instead of c, we are to multiply only by the <ith part of c, 

 then, of course, the product will be only the d th part of 



. ac ac a c 



the former ; that is, it will bo -r- -=-rf, or ,.>.., X , 



o Oft o 



= .. This suggests the rule, which is as follows : 



1 " i 



Rr/LE 1. Multiply the numerators together, tho re- 

 sult will be the numerator of the product. 



2 Multiply the denominators together, and tho result 

 will be tho denominator of the product. 



SOTK. Before performing the operation, see whether either of the frac- 

 tions can be reduced to lower term; if so, reduce tho fraction. And 

 after the operation, see whether a like reduction can be easily made in 

 the result ; if so, reduce accordingly. (See preceding Non). 



It is of importance, in dealing with algebraic frac- 

 tions, that wo always keep in remembrance the factors 

 of such expressions as a 2 b 3 , a* 6*, a 8 fc 8 , <fcc. ; as 

 also of o a + 2ab + b 3 , and a" 2ab + b 3 . You have 

 seen at page 400, that the difference of the square* of two 

 quantities is the product of the sum and difference of tho 

 quantities ; so that a 2 b 3 = (a + b) (a b) ; a 4 6* 

 = (a & J ) (a 2 6; ; o 6 8 = (a 3 + b 3 ) (a 3 b 3 ), tc. ; 

 also that a j + Zab + b* = (a + 6) (a + b), or (a + 6)'; 

 and that a 2 2ab + 6 ! = (a 6) (0 6), or (a i) 2 . As 

 there will bo frequent occasion to apply those truths in 

 operations with fractions, we repeat them here, that you 

 may have full warning of what you will bo expected to 

 remember. 



. 3* 2x 6x* 6x s 



A J\ ~~ " 



2. 



3x x. 



10 

 23? Sx' 



(i\)(x 



Hero it is easy to see that tho 



factor x will enter into both numerator and denominator 

 of tho product : it is useless to allow it thus to enter, 

 and afterwards to expunge it ; we, therefore, silently 

 suppress or cancel this common factor, and so preclude 

 its appearance altogether. In like manner, the factor 5, 

 foreseen to enter numerator and denominator of tho 

 product, is, at the outset, cancelled; so that, before 

 actually multiplying, we imagine tho factors changed 



for X = ' - ^, the product in its 



lowest terms. You are aware that all this is exactly what 

 we should do in common arithmetic : and have, in fact, 

 nothing to learn in the management of algebraic fractions 

 that is not equally necessary in the fractions of pure 

 arithmetic. Suppose the example hero commented upon 



had been *?-= X , "' 



, then, seeing that tho 



6 2x 8 4x 

 second fraction has tho factor 2 common to numerator 

 and denominator, you would here, as in arithmetic, 



Sec (MM, p. 11J, el try. 



