LESSONS IN ALGEBRA. 



275 



LESSONS IN ALGEBRA. IX. 



FRACTIONS. 



117. FRACTIONS in algebra, as well as in arithmetic, have 

 reference to parts of numbers or quantities. The term in 

 1 from the Latin word fractio, which Miymflaa * breaking 

 into parts. 



Thus, I is Jaj ^ is J6 ; ^ is fa ; and ~ ia fr. 



Kxprossions in tho form of fractions occur more frequently 

 in algebra than in arithmetic. Indeed, thu numerator of every 

 fraction may be considered as a dividend, of which the denomi- 

 nator is a tliuisor. 



1 1:. Tho value of a fraction is the quotient of tho numerator 



divided by the denominator. Thus, tho value of - is 3 ; the 





ao 



value of -5- is o ; and tho value of 

 o 



aa bb 



a 6 



, is a .+ b. 



120. From this it is evident, that whatever changes are made 

 in tho terms of a fraction, if the quotients be not altered, the 

 value of the fraction remains the same. For any fraction, 

 therefore, we may substitute any other fraction which will give 

 the same quotient. 



4 10 46a 8drx 6 + 2 

 '' 2 = 5" = 26a = 4dr^ = 3 +T et ' ; f r ^ qu tient " 





each of these instances is 2. 



1 - 1 . It is also evident, from the preceding articles, that if the 

 'lor and denominator be both multiplied, or both divided, 

 by the same quantity, the value of the fraction will not be altered. 

 Thug, I = g, each term being multiplied by 9 ; and J = = -?, 

 each term being divided by 3, and the result by 3 again. 

 a bx abx 36 Ibx kabx . 



oo r- = T = sr = , r = T~r~ * or tno quotient in each case 

 o ao oo 50 Jj(t6 



is x. 



122. Any integral quantity may, without altering its value, 

 be expressed in the form of a fraction, by making unity or 1 the 

 denominator; or by multiplying the quantity into any proposed 

 denominator, and making the product the numerator of the fraction 



a ab ad + ah Gadh 



required. Thus, a = - = r = -f- = , ; the quotient 

 1 o a + n ban 



ef each of these being a. 



dx + hx 2drr + 2dr 



Alsod+?i = :[ ;andr+l = . 



x 2dr 



ON THE SIGNS OF FRACTIONS. 



123. Each sign in the numerator and denominator of a fraction 

 affects only tho single term to which it is prefixed. The dividing 

 line answers the purpose of a parenthesis or vinculum, namely, 

 to connect the several terms of which tho numerator and de- 

 nominator may each be composed. Tho sign prefixed to it, 

 therefore, affects the whole fraction collectively and every term 

 individually. It shows that the value of the whole fraction, 

 and of course every term, is to be subjected to the operation 

 denoted by the sign. Hence, if the sign before the dividing line 

 be changed from + to , or from to + , the value of the whole 

 fraction is also changed. 



Thus it is plain that the value of j- is a. [Art. 111.] But this 

 will become negative if the sign is prefixed to the fraction. 



Hence, y + ^- = y + a. 



But y = y 

 6 



124. In performing fractional operations there is frequent 

 occasion to remove the denominator of the fraction; also to 

 incorporate a fraction with an integer, or with another fraction. 

 In each of these cases, if the sign is prefixed to the dividing 

 line, the signs of all the terms of the numerator must be changed, aa 

 in Art. 64, where a parenthesis, having the sign before it, is 

 removed. 



Thus b 



ad + ah , , 

 =o d 



b d A; andb- 



ad aA 





Next, if all the signs of all the terms in the numerator of a 

 fraction are changed, the value of the fraction is changed in the 



same manner. Thus, ^ = + a [Art. 101] ; but ~^ = a. 



-=a c; but 



ab + be 



= a + c. 



Again, V all the tigm of all Out term* in ike dmomtoatar of a 

 fraction are changed, the value of the fraction it alto changed. 



Thu, ^= + aj but-? 6 --. 



O 



125. If then the tiyn prefixed to a fraction, or the tig** of all 

 the term of the numerator, or the tiyne of all the termt of the 

 denominator, be changed, the value of the fraction will be 

 changed from positive to negative, or from negative to potUiee. 



120. // the tame change be made upon the numerator and 

 denominator oj a fraction at the tame time, they will balance each 

 other, and the value of the fraction will not be altered. Thus, by 



changing the sign of the numerator, tho fraction / = + o 



-ab 



becomes r = o. Bnt by changing the signs of both the nu- 

 merator and the denominator* it becomes == -f- a, where the 



original value is restored. By changing the sign before the f rac- 



ab ab 



tion, the expression y + -r- = y + a becomes y -j- = y a. Bat 



by changing the sign of the numerator also, it becomes 



ab 

 y = where the quotient a is to be subtracted from y, or 



which is th<* same thing [Art. 58], -f a is to be added, making tho 



/ ^ . fi A 



value y + a as at first. In like manner, - =__-^ = 



" 



2 ~ ~~ 2 I 2 2 ~ 



Hence the quotient in division may be set down in different ways 

 and still have the same value. Thus (a c) -4- 6 is either 

 a c a c 



r + ~iT> or i: r 

 b ' b o b 



REDUCTION OF FRACTIONS. 



127. A FRACTION may be reduced to lower terms, by dividing 

 both tlie numerator and denominator by any quantity which will 

 divide them without a remainder; or by throwing out any factor 

 common to both. According to Art. 121, this process will not 

 alter the value of the fractions. 



EXAMPLE. Reduce -r- to lower terms. Ant. - 



CO C 



128. If the same letter or combination of letters is in every 

 term, both of tho numerator and denominator, it may be can- 

 celled, for this is dividing by that letter or combination of 

 letters. [Art. 98.] 



EXAMPLE. Reduce , to lower terms. Ant. 



od+oA d+A 



129. If the numerator and denominator be divided by the 

 greatest common measure, it is evident that tho fraction will be 

 reduced to the lowest terms. 



5a* 

 EXAMPLE. Reduce to its lowest terms. 



_ 5a* Saaaa 5aa 

 Here '3a-* = 3a^=lT An - 



EXERCISE 13. 

 Reduce the following fractions to lower terms : 



l. 



6dm 



7m 

 7mr 



3. 



bo 



(a + be) x m' 

 am + ay 



dky-dj' 



bm+ by* 

 EXERCISE 14. 

 Reduce the following fractions to their lowest terms . 



1. ~ ,. 



Q^S.1 1O/, 



3. 



ft. 



5. ' ~". 

 ax a* 



. 8a-27b 



6ay* 



6av + -lay' 



y 3s* + So* T -' + 2* + 3 

 -&* -5r+l 



10. 



11. 



2* - x.- 1 ' 



18r* - &r + 44* - 5 

 3** + aOr - 57*' -:- 80r - SO* 



l&r* - 53J + 4Sf + f 

 8t* - 3Oi + SL - It 



2ir - 22j - U + 84i- &r 

 Ifei _ is* _ 14t x 3QK1 - Uc 



