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THE GOLDEN RULE ; OR, SINGLE RULE OF THREE
11.This rule, whicb, for its universal use in most parts of the mathenatics, is called the golden rule, is also called a rule of proportion, because the number fought bears a certain proporition to one of the niunbers given. s. It is called the rale of three, because it confifts of three ögiven numbers, from which a fourth number is to be found, which, in the direct rule, bears the fame proportion to the second number as the third does to the firft. 1 Rule. Multiply the fecond and third numbers together, and divide the product by the firft number, and the quotient is the answer foughty dr the fourth mimber.
3d numor. Example i. If 3 yds. of muslin cost 12s. what will 9 yds. cost at that rate?
3) 108 7"?; 21.
Here the fourth number or answer, 35, bears the same proportion to the second number 12, as 9 the third number, bears to the first number 3, that is
, it contains it three times; or it bears the fame proportion to the third number 9 as the second number 12 does to the first number 3, viz. contains it four times. This proportion is called direct proportion; from whence this rule is called the rule of three direct, and is always performed as above. . But when the fourth number bears the same proportion to
the second as the first does to the third, it is then called indirect proportion. Questions of this nature belong to the next rule, called the rule of three inverse, of which hereafter.
In order to know which is the second and third number, it must be noted, that of the three numbers which are in every question in this rule, that number which alks the question must occupy the third place, and is called the third number; and that number which is of the same nature with the fourth number or answer, must be the second number, and consequently the other number must be the first. The second and fourth numbers are therefore always of the fame Aature; as are the firft and third. Thus, in the foregoing example, the number 9 alks the question, for the question is, how much will 9 yards cost? 9 is therefore tliet hird number. 12 is of the same nature with the fourth number, being money ; ir muft therefore possess the second place : and 3, the other number, muft be the first, which is of the same nature with the third, viz. yards.
When either the first or third numbers confift of different denominations, they must both be reduced to the same deno. mination; and when the second number consists of divers denominations, it must also be reduced to the lowest this reduction must be performed before the work can be wrought; and it must be observed, that the fourth number or answer to the work, is always of the same denomination with the second number fo reduced.
The numbers being so redaced, the fecond and third numbers are to be multiplied together, and the product divided by the first, as before directed; and the fourth number or answer must be brought into the proper denominations required by reduction, and if any thing remain after the products of the second and third numbers are divided by ile first, such remainder must be reduced into the next lower denomination, and then divided by the first number, as before; and if any thing still renain, it must be reduced into the next lower denomination (if there be any lower), and divided by the first
number; proceed in this manner till the remainder be brought to the lowest denomination.
Example 2. If 12 gallons of brandy cost 41. 105. what will 120 gallons cost at that rate? It number.
2d number. 3d number, If 12 gallons cost 41. 10s, whiat will 120 gallons cost?
12) 10800 2,0) 90,0
In this example (the numbers being placed as before directed); the second consisting of two denominations, viz. pounds and shillings, it must be reduced to the lowest denomination (shillings), and the product is go fillings: the question will then be, if 12 gallons coft gos. what will 120 coft? I therefore inultiply 120 the third number, by go the second number, and divide the product by 12 the first number, and the quotient goo is the fourth number, or answer to the question, which, because the second number is reduced to fhillings, is shillings also, and is divided by 20 to bring them ipto pounds, and the quotient is 45 pounds, the true answer, or price of 120 gallons at that rate.
The Proof There are several methods of proving questions in the rule of three, but the truest and most improving to the learner is, to back state the question : thus, to prove the last example, I state the question backwards, making that number which was the fourth number in the question the first number in the proof, and that which was the third number here I make the second, and the second I make the third. IN number. 2d number
3d number. Proof. If 451. purchases 120 gallons, what will 4. 105. purchase?
90 ( 900 9,00)108,00
In the proof of this example, I reduce the first number 451. into fhillings, because the third number 4!. 105. must be re. duced into shillings, consisting of pounds and shillings; and then multiplying the second and third numbers together, and dividing by the first, the answer is 12 gallons, as in the example; it therefore proves the work right.
Example 3. If the income of a person be 3 farthings a minute, how much is it per annum?
3d number, Say, it 1 min. produce 3 farthings, what will 365 days 6 hours produce ?
3 4) 1577850 farthings 12) 394470 pence 2,0), 3287,2-6
Answer 16431. 125. 68.
Here the 365 days 6 hours are reduced into minutes by multiplying first by 24 and then by 60, the product is then multiplied by 3, the second number, and the last product is the answer in farthings, which is brought into pounds and Millings by division, and the answer is 16+31. 125. 6d.,
In the foregoing example the firft number is an unit; when this is the case, the work is performed by multiplication, and when the third number is an unit, the work is wrought by division, for ļ neither multiplies nor divides ; questions of this fort, therefore, properly belong to reduction. •
Example 4. If the effects of a bankrupt amount to 27961. 10s, and his debts be 99901. 125. it is requested to know how much he can pay in the pound? Vol. ). z
Say, if 9990l. 125. pay 20s, in the pound, what will 27961. 10..
20 In stating this question, I 199812) 7718600(5 say, if 9990l. 125, the whole
999060 amount of the bankrupt's
119540 debts, will pay zos. in the
199812)143446017 pound, as it certainly will, what
1398684 will 27961. 1os. the net value of
35796 the bankrupt's property, pay? --And the answer is 55. 7d. in
143184 the pound.
In performing this example, the first and third numbers are reduced into shillings, and multiplying the second and third numbers together, and dividing by the first, the quotient is 5, which is of the same denomination with the second number, viz. fhillings; and there is a remainder of 119540 millings, which is reduced into pence, and then divided by the first number as before, and it quotes 7 pence, and there yet remains 35796 pence, which, reduced into farthings, does not contain the divisor onçe; these fence, therefore, remain over and above the ss. and 7d. in the pound which the bankrupt pays.
The foregoing examples will be found sufficient to instruct the learner in the nature and method of working this rule; I shall therefore give a few examples for practice, leaving the operation to be performed by the learner.
Example 5. If 5616. of indigo cost nil. 45. what will ico8lb. cost at that rate ?
Say, if 56lb. coft 2245. what will 19081b. cost ? - Anfuer 40325. or 2012. 125.
Example 6. If a debtor owes his creditors 5931. 125. and compounds at 7s. 6d. in the pound, what will pay his creditors at that rate?