the second as the first does to the third, it is then called indirect proportion. Queftions 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 în every queftion in this rule, that number which asks the question must occupy the third place, and is called the third number; and that number which is of the fame nature with the fourth number or anfwer, must be the fecond number, and confequently the other number must be the first. The second and fourth numbers are therefore always of the fame nature; as are the firft and third. Thus, in the foregoing example, the number 9 asks the queftion, 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, must be the firft, which is of the fame nature with the third, viz. yards.

When either the firft or third numbers confift of different denominations, they muft 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 firft, 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 first, such remainder must be reduced into the next lower denomination, and then divided by the first 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;

5

number; proceed in this manner till the remainder be brought to the loweft denomination.

Example 2. If 12 gallons of brandy coft 4/. 10s. what wilk 120 gallons coft at that rate?

1ft number.

2d number.

3d number.

If 12 gallons coft 4/. 10s. what will 120 gallons coft?

90

12)10800 2,0) 90,0 Anf. £45

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 fhillings: the question will then be, if 12 gallons coft gos. what will 120 coft? I therefore multiply 120 the third number, by 90 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 truest and most improving to the learner is, to back state the question: thus, to prove the laft example, I ftate 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.

ift number.

zd number.

3d number. Proof. If 451. purchases 120 gallons, what will 4. 108. purchase?, 20

90

900 9,00) 108,00

There

In the proof of this example, I reduce the first number 457. into fhillings, because the third number 47. 10s. must be reduced into fhillings, confifting of pounds and fhillings; and then multiplying the second 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?

1ft number.

zd number.

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

24 1466

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 second number, and the last product is the aufwer in farthings, which is brought into pounds and fhillings by divifion, and the answer is 16431. 12s. 6d.

In the foregoing example the firft number is an units when this is the cafe, the work is performed by multiplication, and when the third number is an unit, the work is wrought by division, för 1 neither multiplies nor divides; queftions of this fort, therefore, properly belong to reduc

tion.

Example 4. If the effects of a bankrupt amount to 27961. 10s. and his debts be 999ol. 125. it is requested to know how much he can pay in the pound? Z

VOL. J.

Say,

Say, if 9990l. 12. pay 20s. in the pound, what will 27967. 10s.

In performing this example, the firft and third numbers are reduced into fhillings, and multiplying the fecond and third numbers together, and dividing by the first, the quotient is 5, which is of the fame denomination with the fecond number, viz. fhillings; and there is a remainder of 119540 shillings, which is reduced into pence, and then divided by the firft number as before, and it quotes 7 pence, and there yet remains 35796 pence, which, reduced into farthings, does not contain the divifor once; these pence, therefore, remain over and above the 5. and 7d. in the pound which the bankrupt pays.

The foregoing examples will be found fufficient to instruct the learner in the nature and method of working this rule; I fhall therefore give a few examples for practice, leaving the operation to be performed by the learner.

Example 5. If 5616. of indigo coft 117. 45. what will

1c08lb. coft at that rate?

Say, if: 56b. coft 2245. what will 1008/b. coft ?- Answer 40323. or 2017. 125.

Example 6. If a debtor owes his creditors 5937. 125. and compounds at 7s. 6d. in the pound, what will pay his credi tors at that rate? 1

Say,

Say, if 20s. can be paid by god. what will pay 118725.? -Answer 2221. 125.

Example 7. If 100l. gain 67. interest in 12 months, how much will 340/. gain in the fame time at that rate of interest ? If 100l. gain 67. what will 340l. gain ?-Anfwer 20l. Ss.

Example 8, A draper bought 6 packs of cloth, each pack containing 12 pieces, for which he paid rosol. being 8s. 4d. per ell Flemish: how many yards were there in each piece?

If 100d. purchases 3 qrs. what will 259200d. purchase?— Anfwer 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 ufe of this rule is, from having the rate, value, proportion, produce, intereft, gain or lofs of one or other number of things, to find the rate, value, proporany tion, produce, interest, gain or loss of one or any other num`ber of the fame things in a direct proportion.

SECT. VIII.

OF THE SINGLE RULE OF THREE INVERSE.

THE rule of three inverfe is that which teaches how from three given numbers to find a fourth, which fhall bear the fame rate or proportion to the fecond number as the first does to the third; or the third number bears the same proportion 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 fame 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 first and