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Say, if 205. can be paid by god. what will pay 118725.? - Answer 2221. 125.
Example 7. If 100l. gain 61. interest in 12 months, how much will 340l. gain in the same time at that rate of interest?
If 100l. gain 51. what will 3401. gain ? - Answer 201. 8s.
Example 8. A draper bought 6 packs of cloth, each pack containing 12 pieces, for which he paid 10Sol. being 8s. 4d. per ell Flemish : how many yards were there in each piece ?
If 100d. purchases 3 qrs. what will 259200d. purchase:Answer 7776 qrs. which divided by 72, the number of pieces in the whole, it quotes 108 grs. or 27 yards in each piece.
The general use of this rule is, from having the rate, value, proportion, produce, interest, gain or loss of one or any other number of things, to find the rate, value, proportion, produce, interest, gain or loss of one or any other onmber of the same things in a direct proportion,
OF THE SINGLE RULE OP THREE INVERSE.
The rule of three inverse is that which teaches how frona three given numbers to find a fourth, which shall bear the fame rate or proportion to the second number as the first does to the third ; or the third number bears the same pro. portion to the second as the first does to the fourth.
Rule. Multiply the first and fecond numbers together, and divide the product by the third.
The method of placing the numbers in this rule is the same as that of the rule of three direct, and therefore need not be repeated.
In order to discover whether a question belong to the direct or inverse rule, it must be remembered that the firft and
third numbers are called the extremes ; if, therefore, the fourth number be greater than the second, the less extreme must be the divisor; but if it be less, the greater extreme must be the divisor.
And when the first number is the divisor, the work belongs to the rule of three direct ; but when the third number is the divisor, it belongs to the rule of three inverse; as in the following examples.
Example 1. If 8 men perform any certain piece of work in 6 days, how many men will perform the fame work in 3 days? If 6 days require 8 men, what will 3 days require?
3)48(16 Answer 16 men. In this example I consider, that if to do the work in 6 days it requires the labour of 8 men, then, to do the same work in 3 days, it will require more men, consequently the fourth number will be greater than the second, therefore I divide the product of the first and second numbers multiplied together by the third number (the leatt extreme), and the quotient is 16 (men) for the answer.
The proof of questions in this rule, as well as the foregoing rule, is performed by back ftating the question; but it must be observed, in all questions in the direct rule, the proof is wrought by the rule of three direct; and in questions of the inverse rule the proof is wrought by the rule of three inverse.
Thus, to prove the foregoing example, I say, if 16 men require 3 days to perform the work, how many days will s men require to perform it? If 16 men require 3 days, what will 8 men require ? 3
Anfwer 6 days. Example 2. If a loaf at a certain price weighs 1 lb. when wheat is 6s. per bufhel, what should it weigh when wheat is 45. 6d. per busbel?
If 72d. ser bushel give 9 half lbs. what will 54d. give?
Anf. 12 half lbs, or 61b.
In this example, I multiply 72 pence, the price of the bushel of wheat, by 9, the number of the half pounds in the loaf, and dividing by 54 pence, the price of the bushel of wheat at the other price, the answer is 12 half pounds or 6 pounds for the weight of the loaf at that price of the , wheat.
In London the price of the loaf is varied, and the weight continues the sanie. Questions which concern the price are wrought by making the price the second number...,
Example 3. If a board be 8 inches broad, how much in length will make a square foot ?
If i2 in. in breadth require 12. i1. in length, what will 8 in. in breadth require?
Anf. 18. Ex.imple 4. How many yards of thalloon at 39rs. wide are fufficient to line throughout the garments made with 1000 yards of cloth at 7qrs. wide ? If 7qrs. wide require 1oooyds, what will 39rs wide require?
3) 7000(2333) Anf. 23331 yards. Qu. 5. If 1501. be lent for nine months, how long should gol. be lent to gain the same interest at the fame rate : Answer 15 months.
Qu. 6. If a colonel be besieged in a town with 1000 men, having provisions for only 2 months, bow many must he dismiss, that the provision inay serve the remainder 5 months ? ---Anficer 603, and retain 400.
21.7. If a person perform a journey. (travelling at an uniform rate) in 24 days, by travelling 12 hours per day,
how long will it take to perform the same journey by iravelling 16 hours per day at the same rate :- Anf. 18 days:
Qu. 8. If a carrier carry 12 cwt. 72 miles for 5l. how many miles will he carry 18 cwt. for the same money ? Answer 48 miles.
This rule serves to find a fourth number to the given three, in an inverted proportion. And it particularly answers feven forts of queftions : viz. 1. Having the value of two different forts of coin, it shews how many pieces of the one are equal in value to a given number of the other. 2. Froin two different values of one commodity, and the value of an. article made from the same commodity at one value thereof, to find the weight, measure, &c. of the fame article; or, on the contrary, from the values of the commodity and the weight, measure, &c. of the article, to find the value thereof (the 2d example is of this nature). 3. From the breadths of two equal rectangular figures, and the length of one of chem, to find the length of the other; or, from the two lengths and one breadth, to find the other breadth (of this nature are the third and fourth examples). 4. From the given weight, expense of carriage, and number of miles carriage of any goods, to find the number of miles any other weight could be carried for the same price : from a given weight and price, and two diftances, to find the weight answerable to the other distance (of this nature is the eighth question). 5. From two sums of money lent, and the time for which one of them is lent, to find the time for which the other fhould be lent'; or, from the two different times, and the fun which is lent for one of them, to find the sum which should be lent for the other time (of this nature is the fifth example). 6. From the quantity of work which a given number of men can perform in a given time, to find the number of men that can perform it in iny other given time; or, from the number of men, to find the time any other given number would require (of this nature is the firft example). 7. From the quantity of provisions, or inoney, and
the number of men, or other creatures, it would ferve a certain time, to find the number of men, or other creatures, it would serve any other time; or, from the quantity of provision and the number of consumers, to find the time it would serve any other number of consumers (of this nature is the fixth example).
OF THE DOUBLE RULE OF THREE DIRECT
In this rule there are five given numbers to find a fixth, which shall bear the same proportion to the product of the fourth and fifth numbers, as the third number bears to the product of the first and fecond.
Questions in this rule are resolved either by two opera-, tions in the fingle rule of three direct, or the rule of three composed of five given numbers.
Each question in this rule confifts of two parts, the suppofition and the demand.
Rule 1. By two operations. Place that number which is of the same nature with the fixth number, or answer, in the second place in the first operation ; and the two other numbers in the suppofition in the first place, the one over the other; and the two numbers in the demand in the third place, one over the other, in like manner as the two in the first place; obferving that the bottom numbers in the first and third places be of like nature, as will also the top ones.
* Some modern writers compound the two double rules of three into one; I have, however, given them diftinétly, being more consopant to the true theory of science. In the next Section is, nerertheless, shewn an infallible method of working both by one rule.