REDUCTION OF FRACTIONS.] 



MATHEMATICS. ALGEBRA. 



4GT 



RULE. Multiply the integral part by the denominator 

 of the fractional part ; connect the product with the 

 numerator, by the sign of that numerator, and place the 

 denominator underneath. 



i^-. Here nothing more is done than 



1. a +f 



ab 



replacing a by , which is, of course, equivalent to it. 



The integral part a is converted into a fractional form, 

 by multiplying it by the given denominator 6, and divid- 

 ing the result by the same letter, thus destroying the 

 eil'uct of the multi] ilication. And this is all that is done 



in operations of this kind. 



4 - 



3. 



b ai(x y) 6 



ab x-\-y 



x y 



x y 



5. ?+*- 



.''i 

 6. *!/)*_ 2 



!/ 



xy 



xy 



"o reduce an improper fraction to a mixed quantity. 

 ULK. Divide the numerator by the denominator, as 

 far as the division can be carried, and to the quotient 

 t the remainder with the denominator underneath. 

 If there be no remainder, the quotient, which will then 

 l>e wholly integral, will be the complete value ; showing 

 that the so-called improper fraction is an integral quan- 

 tised under a fractional form. 



This operation, you see, is nothing but that of divi- 

 sion : the examples at page 4C3, serve as well as any to 

 illustrate the present rule ; but a few others may be 

 here. 



4a 



a' <E a + l 

 a x 



_ 



4a 



._ 



a at 



; _ 7 



r 



KXAMPLES FOR EXERCISE IN THE TWO PRECEDING RULES. 



Prove that the following are IDEXTICAL EQUATIONS : _ 





a x a x 



a x 



5. 



a 2 -{-ax 



f, x 

 o. x 



must be reduced to a common denominator. No two 

 quantities can be actually added together, or subtracted 

 the one from the other, so long as the denominations of 

 the quantities are different. We cannot add 4 shillings 

 to 2 pounds ; but can only connect them together as 

 distinct quantities, till both are brought to the same 

 denomination. It is only then that they can be actually 

 incorporated in one sum ; 40s. and 4s. make 44s. ; the 2 

 and the 4 make 6; but these are neither pounds nor 

 shillings, and have no meaning in reference to the things 

 proposed. A fraction, like a concrete quantity, denotes 

 a stated number of things of a stated denomination ; the 

 number is expressed by the numerator, the denomination 

 by the denominator; and .*. so long as the denominators 

 are different, two fractions can no more be added or sub- 

 tracted than pounds and shillings. It follows, therefore, 

 that before fractions can be fitted for addition or sub- 

 traction, they must be prepared for these operations by 

 a previous reduction of them to common denominators. 

 The rule for this reduction is given below ; it is founded 

 upon the obvious principle that a fraction is not altered 

 in value, though the numerator be multiplied by any 

 quantity whatever, provided only that the denominator 

 be multiplied by the same thing ; since the new factor in 

 the denominator just counteracts the influence of tho 

 same factor in the numerator : a multiplication, and then 

 a division by the same thing, leaves the quantity operated 

 upon, whatever it may be, virtually untouched. 



RULE 1. Multiply each numerator by the product of 

 all the denominators except its own: the results will be 

 the numerators of the changed fractions. 



2. The product of all the denominators will be the 

 denominator common to all the changed fractions. 



This rule will effect the reduction of fractions to a 

 common denominator in all cases ; but sometimes the 

 desired change may be brought about in an easier way. 

 The common denominator, found by the rule, is evidently 

 such that each of the proposed denominators is always a 

 factor of it. It is the object of the rule to make sure of 

 such a number in every case; but it often happens that a 

 smaller number exists, such that each denominator is a 

 factor of it ; and whenever such smaller number can bo 

 discovered with little trouble, it is of course bettor to 

 use it, than a larger number. The smallest number 

 possible is called the lead common multiple of the deno- 

 minators ; and a little examination of the denominators 

 will often enable us to arrive at it very readily, as will be 

 seen in some of the following examples : 



j a . b , c ayz , bxz , cxy 



xyz xyz xyz xyz 



Here the three fractions on the right of the sign of 

 equality are respectively the same, in value, as those on, 

 the left ; and the denominator xyz is common to all. 



2 ^ 



2 



2x43 



30 



30 



3x 



-_ 



- - = ^ 



x 2 + a 2 x 2 + a* 



s. ( x +y + '- 





10. 



* _ 3* _ 



_ 



-S _j _ (x y z)(x y 

 / 2xy 



(( I) 1 *) 



z) 



X 2 



2 



4. ' + . Here it is plain that 12 is a num- 

 ii'i. 4 



ber, and the least number, such that 6 and 4 are each a 

 factor of it, /. 12a is the least common multiple of 

 Ca and 4, so that the common denominator need bo no 

 higher than 12a; this 12a is the first denominator above 

 multiplied by 2, /. multiplying the numerator also by 2, 



the changed fraction is ' "*" '. Again, the same 

 J.2& 



12a is the second denominator multiplied by 3a, .'. 

 multiplying the numerator also by 3a, the second 



fraction is ; so that ' 



To reduce fractions to a common denominator. 

 Before fractions can be either added or subtracted they 

 Sec ante, p. Ivi. t See p. 4M. 



changed 



2(2x + 1) , 15ox 

 12a 12o 



would have been twice as great. 

 3 



Ca 



, 



r 



fa. 



common 



P 



5 - 



Here, since wo know that (x -f a) 



