FROPOBTION. ] 



MATHEMATICS. ARITHMETIC. 



417 



that the number, furnished for the product, is the same 

 in one case as the other ; this number, therefore, re- 

 presents so many pence ; which, when reduced to pounds, 

 ia 4240. 



The following are a few examples for exercise : 



(1.) If 57 cwt. cost 216, what will 95 cwt. cost? 

 Ans. 360. 



(2.) If 148 gallons cost 119 10s., how many gallons' 

 will 89 12s 6d. buy ? Ans. 111. 



(3.) What is the value of 2 qr. 24 Ib. at 5 7s. 4<Z. per 

 cwt. 1 Ans. 3 16s. Sd. 



(4.) What is the income of a person who pays 

 22 7s. od. for income tax, at the rate of Id. in the 

 pound? Ans "i>7. 



(5.) 44 j guineas used to be coined out of 1 Ib. of 

 standard gold : how many sovereigns are now coined out 

 of this weight? Ans. 46; f. 



(6.) 60s. are coined out of 1 Ib. of standard silver: 

 what is gained in coining 100 of silver, if the price of 

 the silver be 5s 2d. per oz. t Ans. 6 9 r ' r . 



The Double Bute of Three. The double rule of three 

 is so called because there are at least two single rule-of- 

 throe statings implied in it. The following is an example, 

 namely : 



If 12 horses plough 11 acres in 5 days, how many 

 horses will plough 33 acres in 18 days ? 



Tins may be divided into two single rule-of- three 

 questions ; thus 1st. If 12 horses plough 11 acres, how 

 many will plough 33 acres in ike same time ? 



11 ; 33 ; : 12 horses ; * horses = 36 horses. 



2nd. If 36 horses can perform a work in 5 days, how 

 many can perform the name in 18 days ? By the former 

 rule, 



18 ; 5 : ; 36 horses : X J? horses = 10 horses. 



18 



It is plain that by these two single mle-of-three opera- 

 tions, tliu correct answer to the question is obtained ; 

 but it ig more readily obtained by the following arrange- 

 ment : 



\l ^ } ; ; 12 horses : ] L ' x :>> <-? horses = 10 horses. 



The fourth term of this compound proportion, as it Li 

 called, being got by multiplying the third term by the 

 product of the consequents, and then dividing by the pro- 

 duct of the antecedents ; and it is by the same multipli- 

 cations and divisions that the final result is arrived at in 

 the two distinct statings above. This more compact form 

 of working is described in the following rule : 



Hi I.E. Put for the third term that one of the given 

 quantities which is of the same kind as the quantity 

 sought, just as in the single rule of three. 



Then taking any pair of the remaining quantities like 

 in kind, complete the stating, as if for the single rule of 

 three, paying no regard to the other quantities, or rather 

 considering them to remain the same. 



Then take another pair, like in kind, as a new antece- 

 dent and consequent to be placed under the former pair ; 

 these, with the third term above, completing a second 

 single rule-of-three stating. And proceed in tliis way till 

 all the pairs are used. 



Multiply the third term by the product of all the con- 

 sequents, and divide the result by the product of all the 

 antecedents, and the answer will be obtained. 



Each given antecedent and consequent is of course to 

 be regarded as an abstract number. It is convenient to 

 indicate merely the several multiplications, at first, to 

 place the divisor under the dividend, in the form of a 

 fraction, as in the above example, and then, before per- 

 forming the operations, to expunge factors common to 

 numerator and denominator. 



;i'LK. If 15 12s. pay 16 labourers for 18 days, 

 how many labourers will 35 2s. pay, at the same rate, 

 for 24 days ? 



As the answer is to bo a certain number of labourers, 

 the given 16 labourers is to be the third term ; then 

 taking day* for the first antecedent and consequent, and 

 money for the second antecedent and consequent, attend- 



ing to whether either consequent should be greater or 

 less than its antecedent, as in the former rule, the 

 operation is as follows : 



Jt?15 12s. ^ 3 8 5 2s. } ' ' 161ab urers : 27 labourers. 

 20 20 



312 



702 therefore 



4x3x351 



= 1^ ^ lab. = 2fi lab. = 27 lab. 



351 

 13 



___ 



24 x 312 156 



The 18 is placed in the second term, because fewer 

 labourers are required for 24 days, the work being the 

 same, than for 18 ; and the 35 2s. is placed in the 

 second term, because mpre labourers can be paid for that 

 sum than for 15 12s., the time being the same. If the 

 question had been worked by two single rule-of-three 

 statings, we should have had 



1st. 24 ; 18 : : 16 lab. : X kb. = 12. 



2nd. 312 : 702 lab. 



2i 



You see, therefore, that the double rule of three merely 

 compounds the several single proportions into one ; it is 

 thus called compound proportion. 



We add two examples for exercise in this rule : 



1. If 8 persons can be boarded for 16 weeks for 42, 

 how long will 100 support 6 persons at the same rate ( 

 Ans. 50$$ weeks. 



2. If a family of 13 persons spend 64 in butcher's 

 meat in 8 months, when meat is (W. per Ib., how much 

 money, at the same rate of consumption, should a family 

 of 12 persons spend in 9 mouths, when meat is (j'jd. 

 per Ib. Ans. 72. 



In this example, there would be three separate statings, 

 if the question were worked by the single rule of three ; 

 these are here to be compounded into one. 



Decimals. It was observed at the commencement of 

 this chapter, that in our system of arithmetic, numbers 

 are expressed in the decimal notation, and the reason for 

 this designation was stated: it is simply this namely, 

 that the unit of any figure in a number has a value of 

 ten times the unit of the figure in the next place to the 

 right. Thus, in a number consisting of unit-figures as for 

 instance, in the number 1111 the second unit, beginning 

 with the right-hand one, is 10 times the first, the third 

 10 times the second, the fourth 10 times the third, and 

 so on ; or beginning with the first on the left, the second 

 is the tenth part of the first, the third the tenth part of 

 the second, and so on till we come down to the last unit, 

 which is merely (me. Now, we may evidently extend this 

 principle still further ; and, on the same plan, may repre- 

 sent one-tenth of one, one-tenth of this, or one-hundredth 

 of one, one-thousandth of one, and so on, by simply 

 putting some mark of separation between the integers ami 

 these fractions. The mark actually used is a dot, thus : 

 1111-1111. The unit next the dot, on the left, is 1 ; tin 

 unit one place from this on the left is 10 ; the next i.s 

 100 ; the next, 1000 ; and so on. In like manner, tho 

 unit one place from the 1 on the right, is ^, the next 

 ffaf, the next -^faq, and so on. This being agreed upon, 

 it is easy to interpret such a number as 3t>'427: it is 

 36 + A 4" T$Tf ~F iViHr > eacn fig ure , t tuo right of the 

 point, being a fraction of a known denominator ; the de- 

 nominator b<ung 10 for the first figure, 100 for tlie second, 

 1000 for the third, and so on. The sum of the fractions 

 represented by the decimal '427, above, is obviously $$?$ ; 

 in like manner, the fraction expressed by '2643 is ft ., 

 and in general the denominator of the equivalent fraction 

 is always 1 followed by as many zeros as there are decimal 

 places, the numerator being the number itself, when the 

 prefixed dot, or drcinial point, as it ia called, is sup- 

 pressed. You will thus easily see that tho following are 

 so many identities namely : 

 2i-G=2l&; 136'44=lMi(toi 7.'! -01 l=7.T ,1, 4 ,,\,; 2-07=2 T !ta 



Any decimal may therefore bo converted into its equi- 

 valent fraction at sight : it will be shown presently hovr 

 any fraction may be converted into its equivalent decimal, 

 though not with the same rapidity. 



