ALGEBRA. 



minator of each fraction by the denomi- 

 nators of all the rest, divided respective- 

 ly by their greatest common measure ; 

 and "the fractions will be reduced to a 

 common denominator, in lower terms 

 than they would have been by proceed- 

 ing according to the former rule. 



Thus. --. - --, 



m x my m z 



mon denominator are 



reduced to a com- 



a V z b x z 

 - ; - 

 m x i/ z m x y z ; 



bd 



c d a ') c d a b c d 



Ex.3, a ;-= 7 r-= ; 



b b b b 



Ex. 



a c a d b c b d 



a c a d 



~b cb d~ 



b c+bd 

 'bcbd 



b cb d 



The sign of b d is negative, because 

 every part of the latter fraction is to be 

 taken from the former. 



ON THE ADDITION AND SUBTRACTION OF 

 FRACTIONS. 



If the fractions to be added have a com- 

 mon denominator, their sum is found by add- 

 ing the numerators together, and retaining 

 the common denominator. Thus, 



a c -f-c 



Tf the fractions have not a common de- 

 nominator, they must be transformed to 

 others of the same value, which have a 

 common denominator, and then the addi- 

 tion may take place as before. 



ad bc 



ad+bc 

 '' bd ' 



Ex - 3 -^n- 



_ a 6-fo-H 





\ b a^b 1 a* I 



Here 



a is considered as a fraction whose deno- 

 minator is unity. 



If tivo fractions have a common denomi- 

 nator, their difference is found by taking the 

 difference of the numerators, and retaining 

 the common denominator. Thus, 



c a c 



~b b~' 



I 



If they have not a common denomina- 

 tor, they must be transformed to others 

 of the same value which have a common 

 denominator, and then the subtraction 

 may take place as above. 



ON THE MULTIPL1C ATION AND DIVISION 

 OF FRACTIONS. 



To multiply a fraction by any quantity, 

 multiply the numerator by that quantity, and 

 retain the denominator. 



Thus, - x c = -T-. For if the quantity 



to be divided be c times as great as be- 

 fore, and the divisor the same, the quo- 

 tient must be c times as great. 



The product of two fractions is found by 

 multiplying the numerators together for a new 

 numerator, and the denominators for a new 

 denominator. 



Let - and - be the two fractions ; then 

 b d 



a c 

 '"bd' 



For if -7= oc and = y, by 



multiplying the equal quantities and x, 



by b, a=6 x ,- in the same manner, cdy,- 

 therefore a c = b d x y,- dividing these 

 equal quantities, a c and b d x y, by b d, 



To divide a fraction by any quantity, 

 multiply the denominator by that quantity, 

 and retain the numerator. 



The fraction? divided by c, is. Be- 

 b be 



a a c , .. . a 



cause =7 , and a c th part of this js ; 



b b c be 



the quantity to be divided being a c 1 ^ 

 part of what it was before, and the divi- 

 sor the same. 



The result is the same, whether the de- 

 nominator is multiplied by the quantity, 

 or the numerator divided by it. 



Let the fraction be -r-t; if the denomi- 

 bd 



