444 



MATHEMATICS. ARITHMETIC 



[ADDITION or FRACTIONS. 



i 



pomaiftn rtrnuiiiinnlnr, thry mn neither be add' .1 to n..r 



ictrd from one anothrr : the reduction nf fra. 

 to a common denominator U thus a preliminary, indis- 

 pensably necessary, for the application to them of the 

 rules of arithmetic. The o|H-ratim is also useful in 

 i to discover at a glance which of t 

 newly equal, is really th greater. Thus, of 

 the two fractions, i'., it, we see in a moment which is 

 the greater, became their denominator* are the same ; but 

 we could not so readily and confidently state which is the 

 greater of { and ,, ; i inner are only these re- 



dooed to a common denominator the values are the same. 

 The rule for reducing fractions to a common denomi- 

 nator ii as follows : 



: ultiply eae\ numerator by the product of 

 .nominators of all the other fractions ; we shall thus 

 get the numerator* of the changed fractious. 



Multiply all the denominator* together ; the product 

 will be the common denominator belonging to each 

 changed numerator. 



For example: in order to reduce the fractions J, \, j, 

 to others of the same value with the same denominator, 

 we proceed as follows : 



2X6X7- 701 



1x3x7= 21 Uhe new numerators ; 



3X3X5= 4.1 1 



3x6x7= 105 the common denominator ; 



therefore the equivalent fractions, changed in form as 

 required, are ff f , -fts* iVr- 



i compare these with the original fractions, yon 

 will see that they each arise from multiplying the nu- 

 merator and denominator of the former by the same 



TO 2 x 35 

 number. Thus, io53 y 35 ' an< * ** ** obvious that one 



number divided by another (in this rase 2 by 3), is the 

 same as 35 times the former divided by 35 times the 

 hitter, or any number of times the former divided by the 

 same number of times the latter. If you have any doubt 

 of this, just consider, if yon had to divide 2s. among 3 

 people, whether the share of each would not be the same 

 as if you had to divide 35 times 2s. that is, 70s. among 

 35 times 3 people that is, 105 people. It is plain that, 

 in either case, each would get a third part of 2*., or two- 

 thirds of Is ; or, to view the matter more generally, it 

 is self-evident, that if you multiply any quantity by a 

 number, and then divide by the same number, you vir- 

 tually leave the quantity, as to ralw, untouched ; for 

 multiplication and division by the same number, are 

 two operations which mutually neutralise one another. 

 ,:iy. therefore, always multiply numerator and 

 denominator of a fraction by any number, without 

 changing the value of the fraction. 



The rule just given will ahmys effect the object pro- 

 posed by it ; but not always in the shortest way. In 

 particular cases it will be desirable to proceed dili'erently. 

 Thus, if the fractions J, *, $, are to be changed into 

 equivalent ones with a common denominator, you see, by 

 looking at the denominators, that the thing may be 

 brought about without interfering with the middle frac- 

 tion at all: you have only to multiply the terms of the 

 first fraction by 2, and those of the third by 3, to get the 

 desired result the changed fractions being found in this 

 way to be i had applied the rule, tho new 



fractions would have been <U, JV, If, forms far less 

 dimple than those nlxive, although the same in valve ; 

 they wonld be got by multiplying the terms of the sim- 

 pler fractions, each by 6. In bringing fractions to a 

 common denominator, yon should always be on the look- 

 out for the simplest multiplier of the terms of each that 

 will accomplish the object, and use the rule only as mat- 

 ter of necessity that is, only when simpler multi- 

 pliers than the rule supplies, do not present themselves. 

 Suppose you had }, j, j, do yon not see, from a glance 

 at the denominators, that if the first be multiplied by 3, 

 the second by 4, and tho third by 8, that the products 

 will be all alike ? Multiply, then, tho terms of the first 

 fraction by 3, those of the 'second by 4, and those of the 

 third by 8, and you will got the following viz. , ^, jj, jj, 



for equivalent fractions with a common denominator. The 

 .11 mid have given you the-e ,'',',, {;',', ^'J, which, 

 although equal to, are far less sin 



The smallrst number that can boa common denomi- 

 nator of a row of fractions is evidently the smallest num- 

 ber that is divisible by each of the given denominators : 

 it is called the least common m I'MOM- denomi- 



-. There is a rule for finding the least common 

 multiple of a set of numbers ; but you see that it may 

 often be discovered, without any rule., by a little reflec- 

 tion. \\ e shall give you but one more instance here, 

 since the reduction of fractions to a common denominator, 

 aa observed above, will form a necessary ; n for 



addition and subtraction. Let the fractions be , {,},}: 

 here you see that the first two are brought to a common 

 denominator by merely multiplying the terms of the lirst 

 by 3 ; so that these two fractions are J, f. Again, the 

 List two are brought to a common denominator by merely 

 multiplying the terms of the fourth by 3 ; so that these 

 two fractions are J, J. We have, therefore, now only to 

 find tho /east number which will divide by and 6 ; and 

 it requires but little thought to discover that 18 is that 

 number ; so that we reach the desired result by the fol- 

 lowing steps : 



*.> {> J 



or, 8, J, , | 

 or, U, IS, A, H 



where yon see that a fraction in either of the lower 

 rows is merely that above, with its terms multiplied by 

 the same number. If you had applied the rule to the 

 row of fractions, you would have got3x9x6x2= 

 o-4 for a common denominator, instead of the more 

 simple number 18. 



Addition and Si&trartinn of Fraction!. RtTLE. Re- 

 duce the fractions to equivalent ones, having a common 

 denominator : then add or subtract the numerators, as pro- 

 posed, and put the common denominator under the result. 



For instance, let the fractions be J and \ : these, re. 

 duced to a common denominator, are J J and J, 8 ,, there- 

 fore J +>;=,",, their stun: ,' ; ' }||, their il, if < rence. 



Again, let it be required to find the value of ; -(- *. 

 Here the second and third fractions will have a common 

 initiator, if the terms of the third are multiplied by 

 3: the differing denominators will then be 5 and it; that 

 is, we shall have J-f-J 1 = 1 J; that is, by tho rule 

 for the common denominator, j J ' = ? 



Suppose the value of 1 -f s & were required. It 

 is easy to see that the denominators will be made alike 

 if the first be multiplied by 4 and the second by 3 ; so 

 that, multiplying numerators, as well as denominators, 



we have i - M j-j = i > ;; - ., v 



And in like manner are the results following obtained : 



In the subtraction of mixed quantities, it sometimes 

 happens that the fractional part of the subtracted quan- 

 tity is greater than the fractional part of that from which 

 it is to be taken : when this is the case, it is better to 

 convert a unit of the latter into a fraction, and to in- 

 corporate it with the fractional part : we shall thus have 

 an improper fraction, from which the tubtraetive fraction 

 may bo taken ; thus, if wo had to take 3J from 65, we 

 see, when the fractions are brought to a common ttl 

 initiator, by multiplying the terms of the first by 4, and 

 those of the second by 3, that tho subtractivo fraction 

 is greater than ^-, the other fraction ; wo thei 



of the 5, considering 5 to be -I-!], and 



ractionise a unit 

 therefore i>J, or 5A, > 

 thus: 5 3f-6& 



hat the work stands 



- --:: 



Mullijilicatwn and Division, of Fractiont. Afij'; 

 cation. If we have to multiply a fraction by a whole 

 numlicr, the product will, of course, be as many times 

 that fraction as there are units in tho whole number : 

 thus, ^x3 = J; that is, 3 times two-jlf/lm: the tit-nomi- 

 nator is not operated upon, because this merely states 

 the denominations, not the HHI/I'U c of them. If, in 

 of 3, the multiplier had been only a fourth part of 3, 

 that is J, then only a fourth part of the above product, 



