In the 11th example, I multiply the 350 ducats by 50,.to bring them into pence, and then divide by 12, and it quotes 1458 fhillings and 4 pence: I then divide the fhillings by 20, and the quotient is 724. 185, 4d. for the answer.

In the 12th example the 2461. 18s. 6d. Flemish money is reduced into pence, and then divided by 366, the course of exchange; and the answer is 1617. 185. 4‡d. and 114 farthings remain.

The 13th example is wrought in the fame manner as the 11th: viz. by reducing the pieces of eight into pence; for the fraction of a penny, I multiply the given number - of pieces of eight by 5, the numerator, or upper figure of the fraction, and divide the product by 8, the denominator, or - lower figure of the fraction, and to the quotient I add the pence contained in the pieces of eight, and then reduce the whole into fhillings and pounds, by dividing by 12 and 20, as before, and the answer is 9137. 185. 4d

This method of reducing foreign coin into English may ferve for those who are unacquainted with the rule of practice; for practice performs this much more expeditiously, as will be fhewn in its proper place.

Thofe fums in reduction, in which both divifion and multiplication are used, must be proved by multiplication and divifion; as, for example, that part which is performed by multiplication must be proved by divifion, and that part performed by divifion must be proved by multiplication.

The foregoing examples, perfectly understood, will be fufficient to give the learner a complete knowledge of this rule, and the various ufes to which it may be applied.

It is abfolutely neceffary that the learner be perfectly acquainted with what has been delivered in the foregoing part of this chapter, as all the following rules in arithmetic are performed by one or more of the foregoing rules; I have therefore been more explicit in the former part; being the bafis of the whole.

SECT.

THE GOLDEN RULE; Or, single rule of thRER DIRECT.

THIS rule, which, for its univerfal ufe in moft parts of the mathematics, is called the golden rule, is alfo called a rule of sproportion, because the number fought bears a certain proportion to one of the numbers given.

It is called the rale of three, because it confifts of three ogiven numbers, from which a fourth number is to be found, which, in the direct rule, bears the fame proportion to the fecond number as the third does to the firft.

Rule. Multiply the fecond and third numbers together, -and divide the product by the firft number, and the quotient is the answer fought, dr the fourth mimber.

ift number..

2d number.

3d number.

Example 1. If 3 yds. of muffin coft 125, what will 9 yds.

Here the fourth number or anfwer, 35, bears the fame proportion to the fecond 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 fecond 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 fame proportion to

the

the second as the firft does to the third, it is then called indire proportion. Questions of this nature belong to the next rule, called the rule of three inverfe, of which hereafter.

In order to know which is the fecond and third number, it must be noted, that of the three numbers which are in every question in this rule, that number which afks the question muft occupy the third place, and is called the third number; and that number which is of the fame nature with the fourth number or answer, muft be the fecond number, and confequently the other number must be the firft. The fecond and fourth numbers are therefore always of the fame nature; as are the first and third. Thus, in the foregoing example, the number 9 asks the question, for the question is, how much will 9 yards coft? 9 is therefore the third number. 12 is of the fame nature with the fourth number, being money; it muft therefore poffefs the fecond place: and 3, the other number, muft be the first, which is of the fame nature with the third, viz. yards.

When either the first or third numbers confist of different denominations, they must both be reduced to the fame denomination; and when the fecond number confifts of divers denominations, it must alfo be reduced to the loweft:this reduction must be performed before the work can be wrought and it must be obferved, that the fourth number or answer to the work, is always of the fame denomination with the fecond number fo reduced.

The numbers being fo reduced, 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 fecond and third numbers are divided by the firft, fuch remainder must be reduced into the next lower denomination, and then divided by the firft number, as before; and if any thing ftill remain, 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 coft 4/. 10s. what will 120 gallons coft at that rate?

3d number.

If 12 gallons coft 41. 10s. what will 120 gallons coft?

In this example (the numbers being placed as before directed); the second confifting of two denominations, viz. pounds and fhillings, it must be reduced to the lowest denomination (fhillings), and the product is 90 fuillings: the queftion will then be, if 12 gallons coft gos. what will 120 coft? I therefore multiply 120 the third number, by go the fecond number, and divide the product by 12 the first number, and the quotient 900 is the fourth number, or answer to the question, which, because the second number is reduced to fhillings, is fhillings alfo, and is divided by 20 to bring them into 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 trueft and most improving to the learner is, to back state the queftion: 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 fecond, and the fecond I make the third.

A number.

2d number.

3d number. Proof. If 454. purchafes 120 gallons, what will 41. 10s. purchase?

20

90

900 9,00) 108,00

There

In the proof of this example, I reduce the first number 451. into fhillings, because the third number 4. 10s. must be reduced into fhillings, confifting of pounds and fhillings; and then multiplying the fecond and third numbers together, and dividing by the firft, the answer is 12 gallons, as in the example; it therefore proves the work right.

Example 3. If the income of a perfon be 3 farthings a minute, how much is it per annum ?

ft number. zd number.

3d number. Say, if 1 min. produce 3 farthings, what will 365 days 6 hours produce?

24

Here the 365 days 6 hours are reduced into minutes by multiplying firft by 24 and then by 60, the product is then multiplied by 3, the fecond number, and the laft product is the answer in farthings, which is brought into pounds and fhillings by divifion, and the anfer is 16437. 12s. 6d.

In the foregoing example the first number is an unit; when this is the cafe, the work is performed by multiplication, and when the third number is an unit, the work is wrought by divifion, fort neither multiplies nor divides; queftions of this fort, therefore, properly belong to reduction.

Example 4. If the effects of a bankrupt amount to 27961. 10s. and his debts be 99907. 125. it is requested to know how much he can pay in the pound?

VOL. I.

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Say,