Ml 



MATHEMATICS-ARITHMETIC. 



[PROPOKTKIX. 



OM bjr the other, and thin get the abstract number 

 here spoken of, till the niuntitied are minced to tlio 



a the Mine in both 



6 6 cwt 



When, however, thu u 

 yon may entirely disregard tUttomination, and oom.ider 

 the resulting number* to be ahtlract number* ; fur the 



thu* ... { U the abstract 



6 yd. 

 o r: 



Thin, though a very nhuom, ii yet 1111 important fact; 

 as hen tlio first and second terms 



of a pro|xirtion arv r.-'lucod to the same denomination, 

 ire may consider both numerator and denominator of 



Second term 

 the fraction y irat ^m, * abstract numbers, as well as 



the fraction iteolf ; and therefore that, without violating 

 any principle of arithmetic, we may write the above 

 equality, namely 

 Second term 



- X Third term Fourth, term 

 First term * 



in the form 



Second term X Third term 



- Fourth term 

 First term 



which is ofti'n the more convenient in practice ; that is, 

 after the reduction to the same denomination, as spoken 

 of above, we may multiply the third term by the second 

 (regarded as a number), and divide the product by the 

 niimlifr denoting the first term. 



\\ , have thus established the general principle of what 

 U called the Rule of Three, the object of which is, when 

 three terms of a proportion ore given, to find the fourth. 



Jtule of Three. 1. Write the three given terms in a 

 row, taking care that the third term is a quantity of the 

 same kind as the required fourth term ; and also that, 

 according as this fourth term U to be greater or less than 

 the third, so may the uxond of the given terms be greater 

 or less than the first. This is called stating the question. 



2. Having thus properly stated the question, bring 

 the first and second terms to the same denomination ; 

 and regard the results as abstract numbers, the deiKimina- 

 tion being suppressed. 



3. Multiply tho third term by the second, which is 

 no>w a number; and divide the product by the first, 

 which is also a number : the quotient will be the required 

 fourth term, in the same denomination, of course, as 

 that in which the third term was used. 



You may sometimes simplify this operation : for the 

 first and second terms may each be divisible by the same 

 number; in which case you may employ only the 

 quotients instead of the quantities themselves, on the 

 principle that a fraction is not altered in value by dis- 

 carding factors common to numerator and denominator. 

 You may, also, in like manner, divide the first and third 

 terms, when possible, on the same principle; and wo 

 would recommend you, in rulc-of-three operations, to be 

 always on the look-out for these means of simplification. 



We shall now show the application of the rule by some 

 examples. 



1. If 16 cwt cost 42 8*., how 



16:26 

 8:13 



42 8: 

 20 



I is IS 



much will 26 cwt. cost ? 



As tho answer or fourth term 

 of the proportion is to be money, 

 the 42 Sn given in the question 

 must be tho third term, 



A* the greater the number of 

 cwt the greater will be the cost, 

 we must arrange the first two 

 tenns of the proportion, so that 

 the second may be the gi 

 consequently, the stating will I.e. 

 16 cwt. : 26 cwt :: 42 8*. : tho 

 required fourth term. But as 

 the first and second terms r.ro 

 already in the same denomina- 

 tion, namely, cwt, no reduction of them U ueccs- 



Iw mml,, p. 44J. 



B4B 

 13 



2544 



MS 



8)11 024. 

 2,0)137,8*. 

 08 184. 





 :42 



t, 

 8 



a 



13 



18*. 



Hero 



Ftrmnt. 



48 



nary, we therefore, in the stating, entirely dis- 

 regard this common denomination, and insert merely the 

 abstract numbers, 16 and 26, as in the margin. But, a 

 glance at these two number* shows that each ix di\ 



; we, therefore, replace them by tin 1 quotients 8 and 

 end it now only remains to multiply -12 8.1. 1 

 nnmUr 13, and to divide the product by 8, to get the 

 sum of money required. For convenience, we reduce 

 the given sum to shillings ; thus bring out tho answer 

 or required fourth term in shillings, and th>n comeit 

 these into pounds. When the work is finished, as 

 here annexed, wo complete the pro|>rtion, in the 

 stating, by putting the result, viz., 08 18*. for tho 

 fourth term. 



You see that, by dividing the first and second terms 

 in the stating by 2, wo have effected a little saving of 

 figures in the work : but we might have simplified 

 further, and have shortened tho operation still more, by 

 dividing the third term and thu reduced first term both 

 by 8 ; so that, having regard to the 

 utmost attainable simplicity, we 

 should have proceeded with the pro- 16 : 20 

 liminary statings as here annexed, 8 : 13 



and thus have reduced the subsequent 1 : 13 



work to the small amount of figures 

 here shown ; 13 times G. is 7 

 that is, 3 J8. : putting down the 

 18s., and carrying the 3, we have 

 13 times 5 =65 ; and the 3 carried makes 68. 

 is another example. 



2. If 28 persons reap a harvest in 36 days, how many 

 will be required to reap it in 21 days ? 



As the answer is to bo number of persons, tho 28 

 persons given in tho question 

 must be the third term of the 

 proportion; and as the fewer 

 the days tho greater must be 

 the number of workmen, wo 

 arrange the first two terms of 

 the proportion so that the 

 second may be the greater; 

 the stating is, therefore, 21 

 days : 36 days : : 28 persons 

 required. 



But, as the first two terms are in tho same denomina- 

 tion, we suppress denomination, and use only the 

 abstract numbers 21 : 36. These we see have a common 

 divisor, 3; we therefore replace them by the quot . 

 7 : 12; but the 7, and the 28 in the thinl term will 

 divide by 7; we thus get the stating in tlio simple form 

 1 ; 12 ; ; 4 persons ; and then proceed as in the margin. 



One example more must suffice. 



A mass of 106 Ib. of Australian gold, sold at tho rate 

 of 3 6s. 8d. per ounoe : how much did it fetch? 



Stating the question as in the margin, putting the 

 greater weight in the 



I term, be- ./. 



cause the greater 1 oz. : 100 Ib. ; ; 3 6 8 ; 4240 

 sum must be in tho 12 20 



fourth, wo see that 



the first and second 1272 66 



terms differ in de- 800 12 



nomination; we 



must, therefore, re- 12)1017600 800 



duce the second to 

 ounces, before we 2,0)8480,0 



can regard the 

 stating in the pr. . 4240 



form for working 

 with : it then becomes 1 oz. ; 1272 07, ; ; 3 6.1. 8d. 



Tho common ilenoniinat ion oUMCl is n<>a suppressed, 

 and, for convenience, the money is reduced to /.. ,u-f, tho 

 denomination in which the required fourth term must 

 therefore appear: we have then merely to multiply 800 

 by tho alistract nuimVr 1-7-'; and tin- required 

 value comes out 1017600 pence ; for, as tho first term U 

 1, thoro is no division. 



Instead of making 1272 tho multiplier of 800, we make 

 800 tho multiplier of 1272, for convenience ; as we know 



Pfron. 



21: 30:: 28 

 7: I'-' 



1: 12:: 4 

 12 



48 persons. 

 the number of persons 



