MULTIPLICATION, ETC., OF FRACTIONS.] M AT H EM ATICS. ABITHMET 1C. 



445 



{, could have resulted. A fourth part of f is ?%, because 

 a number (in this instance 6) divided first \>y 5, and 

 then again by 4, gives the same result as a single division 

 of it by 20. We know, therefore, that f Xf =*V tuat 

 is, it is the product of the numerators, divided by the 

 product of the denominators. 



The same reasoning evidently applies, whatever be 

 the fractions to bo multiplied together : the product of 

 the numerators, divided by the product of the denomi- 

 nators, is the product of the fractions. 



Division. Let us now try to discover how fractions 

 are to be divided ; and first let us consider the case in 

 which the divisor, like the multiplier, in the foregoing 

 instance, is a whole number. 



If we have to perform the operation f-4-3, we see that 

 the quotient of 2 by 5 is to be divided by 3 : but this we 

 know, from simple division, is the same as the quotient 

 of 2 by 15 ; for, if anything be divided by 5 and 

 then the quotient by 3, the result is the same as we 

 should get by dividing that thing at once by 5x3 or 15 ; 

 therefore, 1-5-3=,^. 



But if, instead of 3, the divisor were }, that is, only a 

 fourth part of 3, it is plain that the quotient ought to be 

 4 times as great ; namely, A ; so that g-:-J=^ ; the same 

 result as we should get by inverting the divisor J, and 

 multiplying, instead of dividing. ; for fXt=^- And as 

 the reasoning evidently applies, whatever be the fractions 

 chosen, we infer that division of fractions may always be 

 converted into multiplication by inverting the terms of 

 the divisor ; that is, in fact, by turning the divisor 

 upside down. Thus, <H-?=f Xj=H- ^e are thus 

 led to the following rules, viz. : 



RULE. Fi r inn! i', plication. Multiply the numerators 

 together for the numerator of the product. 



Multiply the denominators together for the denomi- 

 nator of the product. 



Ki LK. For division. Invert the terms of the divisor, 

 and then proceed as in multiplication. 



It is desirable that fractions which appear in the results 

 of any operations, should be reduced to their lowest 

 terms ; that is, that both numerator and denominator 

 should be divided by whatever number vntt divide them. 

 Thus, in the instance above, where the product of and 

 } was found to be $j, the result should be freed from the 

 factor 2, common to both numerator and denominator ; 

 that is, we should divide both terms by 2, and write the 

 :t in the more simple form, ^. The division of 

 both terras by the same number cannot alter the value of 

 the fraction, otherwise the mnlfi/,ln-'tt;<:n, ,,f the terms of 

 a fraction by the same number would alter its value, 

 which we know to be not the case. 



In multiplication and division, we may often prevent 

 the entrance of these superfluous factors in the result ; 

 and it is of course better to do so than to allow them to 

 enter, and then to remove them ; thus, in multiplying 

 J by J we foresee that, as 2 occurs in a numerator and 4 

 in a denominator, 2 will also occur in both numerator 

 and denominator of the product, unless we previously 

 prevent its entrance : this we should do by regarding the 

 proposed fractions as J, J ; the product of which is -fa. 

 All the factors that enter alike into numerators and 

 denominators, should thus be removed, for you then get 

 your product in the simplest form at once, without 

 being at the trouble to reduce it to lower terms. We 

 shall here give example* of this : 



(1-) iXA-jX^-U- (2.) JXi-W-rV (3) * 



-IrV (5 



In this way the entrance of common factors into the 

 numerator and denominator of each result in the follow- 

 ing examples is to be provided against : 



(l.) '.xtf-H- (2.) H-J-H=if (3-) A-t-iV-9. 

 H-A-2J. 5.) U-H-f-lrV- (6-) txf-L 

 n these examples, the rules are applied to pure frac- 

 tions. If we have to deal with mixed quantities, then 

 we must reduce them to improper fractions before using 

 either rule: thus 3<-=-2! = V -J- y V X -ft = V/ 1 $ J- 



Proportion. Four quantities are said to be in propor- 

 tion when the first divided by the second is the same 



abstract number as the third divided by the fourth : thus, 

 the four numbers, 6, 3, 8, 4 are in proportion, because 

 f =J ; and of any two equal fractions, the four terms are 

 in proportion. 



The quotient which arises from dividing one quantity 

 by another of the same kind, is called the ratio of the 

 former to the latter : thus, the ratio of 6 to 3 is 2, and 

 the ratio of 8 to 4 is 2 ; ratio being only another name 

 for the quotient of two quantities of the same kind. A 

 proportion is thus said to be an equality of ratios : ratio 

 implies two terms ; proportion, four. The first term of 

 a ratio (the dividend) is called the antecedent, the second 

 (the divisor) its consequent. Instead of writing the 

 antecedent as the numerator and denominator of a 

 fraction to express the ratio, the same thing is indie ted 

 by simply putting two dots between them ; thus, 6 : 3 is 

 the sainu as 4 ; and 8 : 4 the same as 5 ; so that the 

 proportion above may be expressed in this way, 6:3= 

 8 : 4. Instead of the sign of equality, it is more common 

 to use four dots, and to write the proportion thus : 6:3:: 

 8 : 4, which is read, 6 is to 3 as 8 is to 4 ; a form of 

 expression intended to imply that 6 is related to 3, in 

 point of magnitude, just as 8 is related to 4. This idea 

 of relative or comparative magnitude, which is essential 

 to the correct notion of proportion, forbids our consider- 

 ing the term ratio in the same unrestricted sense as the 

 term quotient: the two terms are to be regarded as 

 meaning the same thing only when dividend and divisor 

 are quantities of the same kind : ratio is always an 

 abstract number. Now, the name quotient, as we have 

 seen, is applied not only to abstract numbers, but to 

 the concrete quantities that arise from taking the third, 

 fourth, fifth, etc. part of concrete quantities. Be careful 

 to observe this distinction ; and to remember that the 

 ratio of an antecedent to its consequent always has 

 reference to the number of times, and parts of a time, 

 by which the former contains the latter : so that it would 

 be absurd to speak of the ratio of one thing to another 

 of a different kind ; as, for instance, of the ratio of 6 

 to the number 3 ; or of 8 cwt. to 4, and so on. 



It thus appears, that if four quantities, ranged in 

 order as above, form a proportion, the first and second 

 must necessarily be of the same kind ; and also that the 

 third and fourth must be of the same kind : thus the 

 following are proportions ; they express equal ratios, the 

 ratio in each case being the abstract number 2 

 6 : 3 : : 8 cwt. : 4 cwt. ; again, 6 yds. : 3 yds. : : 8 oz : 4 oz. 



From what has now been said, you see that the essen- 

 ti:il condition, and the only condition which four quan- 

 tities must fulfil, in order that they may form a proportion, 

 i.s tliis, namely 



First term Third term 



Second term Fourth term 



each of these fractions being an abstract number. If 

 two fractions are equal, we know that two equal fractions 

 will also result from inverting their terms ; so that from 

 the above we may infer that 



Second term Fourth term 



First term Third term 



These being abstract numbers, we may multiply any 

 quantity by either : the results will, of course, be equal, 

 whichever we take for the multiplier, since they them- 

 selves are equal. Let u* multiply "Third term" by 

 each : the results will be 

 Second term 



X Third term -Fourth term. 



First term 



You can have no doubt about the second result, as you 

 know that divisor, multiplied by quotient, gives dividend; 

 and the fraction above, on the right, denotes the quotient 

 of the division of "Fourth term" by "Third term." 



It appears, then, that in order to find the fourth term 

 of a proportion, when the first three terms are given, we 

 have only to divide the second by the first, and to 

 multiply the third by the abstract number furnished by 

 their quotient. Now, if the first and second terms are 

 concrete quantities, you know that you cannot divide 



