RULE of Three, GOLDEN Rule, or Rrts of is one of the most essential rules of arithmetic ; for the foundation of which see the article PROPORTION. It is called the Rule of Three from having three numbers given to find a fourth ; but, more properly, the Rule of Propor tion, because by it we find a fourth num ber proportional to three given numbers: and because of the necessary and exten sive use of it, it is called the Golden Rule. But to give a definition of it, with regard to numbers of particular and de terminate things, it is the rule by which we find a number of any kind of things, as money, weight, Re. so proportional to a given number of the same things, as another number of the same or different things is to a third number of the last kind of thing. For the four numbers that are proportional must either be all ap plied to one kind of things ; or two of them must be of one kind, and the re maining two of another : because there can be no proportion, and consequently no comparison of quantities of different species : as, for example, of three Ma, lings and four days; or of six men and four yards. All questions that fall under this rule may be distinguished into two kinds: the first contains those wherein it is simply and directly proposed to find a fourth proportional to three given num. bers, taken in a certain order : as if it were proposed to find a sum of money so proportioned to one hundred pounds, as sixty-four pounds ten shillings is to eighe teen pounds six shillings and eight-pence, or as forty pounds eight shillings is to six Inindred weight. The second kind con tains all such questions wherein we are left to discover, from the nature and cir cumstances of the question, that a fourth proportional is sought ; and consequent ly, how the state of the proportion, or comparison of the term, is to be made ; which depends upon a clear understand ing of the nature of the question and pro portion. After the given terms are duly ordered, what • remains to be done is to find a fourth proportional. But to re move all difficulties as much as possible, the whole solution is reduced to the fol lowing general rule, which contains what is necessary for solving such questions, wherein the state of the proportion is given; in order to which it is necessary to premise these observations.
1 in all questions that fall under the following rule there is a supposition and a demand : two of the given numbers con tain a supposition, upon the conditions whereof a demand is made, to which the other given term belongs ; and it is there fbre said to raise the question ; because the number sought has such a connection with it as one of these in the supposition has to the other. For example : if three
yards of cloth cost 41. 108. (here is the supposition) what are 7 yards 3 quarters worth? here's the demand or question rais ed upon 7 yards 3 quarters, and the form er supposition.
2. In the question there will sometimes be a superfluous term ; that is, a term which, though it makes a circumstance in the question, yet it is not concerned in the proportion, because it is equally so in both the supposition and demand. This superfluous term is always known by be ing twice mentioned, either directly, or by some word that refers to it. Example, if three men spend 201 in 10 days, how much, at that rate, will they spend in 25 days ? Here the three men is a superflu ous term, the proportion being among the other three given terms, with the number sought ; so that any number of men may be as well supposed as 3.
Rule 1. The superfluous term (if there is one) being cast out, state the other three terms thus : of the two terms in the supposition, one is like the thing sought (that is, of the same kind of thing the same way applied) ; set that one in the second or middle place ; the other term of the supposition set in the first place, or on the left hand of the middle ; and the term that raises the question, or with which the answer is connected, set in the third place, or on the right hand; and thus the extremes are like one another, and the middle term like the thing sought : also the first and second terms contain the supposition, and the third raises the question; so that the third a- d fourth have the same dependence or c nnection as the first and second. 2. Make all the three terms simple numbers of the low est denominations expressed, so that the extremes be of one name. Then, 3. Re peat the questions from the numbers thus stated and reduced (arguing from the sup. position to the demand), and observe whether the number sought ought to be greater or lesser than the middle term, which the nature of the question, rightly conceived, will and, ingly, multiply the middle term by the greater or lesser extreme, and divide the product by the other, the quote is like the middle term, and is the complete an swer, if there is no remainder ; but if there is, then, 4. Reduce the remainder to the denomination next below that of the middle term, and divide by the same divisor, the quotient is another part of the answer in this new denomination.' And if there is here also a remainder, reduce it to the next denomination, and then divide. Go on thus to the lowest deno mination, where, if there is a remainder, it must be applied fraction.wise to the divisor ; and thus you will have the com plete answer in a simple or mixed num ber.