In refolving this question, I fay, as 1s. 2d. the expenfe of making and exporting the linen, is to 15. 9d. the retail price, fo i 100l. to 150l.; thus there is 50 per cent. profit, which, after deducting 26 per cent. the retail trader's profit, leaves 24 per cent. for the factor.

Qu. 4. A merchant bought 100 gallons of brandy, at 65. per gallon, of which quantity 40 gallons were loft; at what price per gallon must he fell the remainder, that he may gain 10 per cent. profit upon the money it cost him?-Anj 115, per gallon.

SECT. XVII.

OF EQUATION OF PAYMENTS.

EQUATION of payments is that rule whereby is discovered the time to pay at one payment several fums due at different times, so that neither party may sustain any loss.

Rule. Multiply each debt by the time at which it is due, and add all the products together; divide the fum of the products by the fum of all the debts, and the quotient will be the answer, or the equated time to pay the whole.

Example 1. A is indebted to B in the fum of 200l. to be paid as follows: 60l. in 4 months, 40/. in 6 months, and 100l. in 10 months; what is the equated time to pay the whole?

Qu. 2. A owes B 1000l. of which is to be paid in 6 months, in 8 months, and the remainder in 10 months; what is the equated time to pay the whole?-Answer 7 months.

In this example, of roool. multiplied by 6 months produces 2, and multiplied by 8, produces 3, and (the remainder of the 1000l.) multiplied by 10, produces 21; and thefe three products added together gives 7 months for the answer.

And, note, when the fums or times of payment are given in fractions, the fum of the products is not to be divided by the fum of the debts, as it is in whole numbers.

24, 3. A tradefman cwes his creditor 144/.; 44. he pays in ready money; 6ol. is to be paid at the expiration of 6 months, and the remaining 401. at the expiration of 8 months; but the tradefman defiring to have more time for the payment of the last 40%. pays his creditor the 6ol.. due 6 months after, in ready money; how long may he defer paying this laft 40l. to make him amends for this prompt payment?-Anf. 17 months.

In this example, the 441. to be paid in ready money is neglected; but for the 60l. paid 6 months before due, I find by the rule of three what intereft it would gain at any rate per cent. in that time, and then how long the 40%. may be lent for that intereft at the fame rate, which I find is 9 months, and which added to S months, its time of payment, gives 17 months for the answer *.

*This rule, though greatly used by men of bufinefs, is not mathematically exact. The reafon of the rule given by many writers, is, that for the debtor paying a fum before the time it is due, an equal um should be forborn, for as long a time after it is due; but this is a mistake, for by the debtor paying money before it is due, he has the difcount only; but keeping the money after it is due, he gains the intereft, which is greater than the discount.

SECT.

SECT. XVIII.

OF THE RULE OF FALSE, SINGLE AND DOUBLE

(GENERALLY CALLED POSITION).

THIS rule teacheth to answer fuch questions as cannot be refolved by any direct rule iu vulgar arithmetic, and must be performed by falfe or feigned numbers.

Position is either fingle or double.

Single pofition is when the queftion can be refolved by one falfe pofition or fet of feigned numbers, and one operation in the rule of three direct.

Rule. Take any number, and proceed exactly the fame as if it were the true number through all the proportions mentioned in the question.

Then fay (by the rule of three) as the refult of this falfe operation is to any of its parts, fo is the true refult in the queftion to the corresponding part required.

Example 1. A fon asking his father his age, the father aufwered, I am double the age of your eldest brother John, and he is three times the age of your youngest brother Henry, and the fum of all our ages is So years; what is each person's age?

Here I fuppofe Henry's age to be 6, at which fuppofition John's age must be 18, the father's 36, and the sum of these three is 60, whereas it fhould be 80; then I fay, as 60 the fum of the falfe fuppofition, is to So the fum of the true one,

fo is 6 the fuppofed age of Henry to 8 his true age; therefore John's age is 24, and the father's 48, as in the example.

Qu. 2. A afked B how much money he had in his pocket; Banfwered, If you give me 4 guineas of the money in your pocket, I fhall have five times as much as you will then have ; but if, instead of that, I flionld give you 2 guineas of the money in my pocket, you will then have twice as much as I fhall then have: how much money had each ?—Ans. 6 guineas.

Qu. 3. A perfon hired a horfe for 9 days, on the following terms for the firft 3 days he was to pay of the hire for the next 3 days, and for each of the laft 3 days as much as the hire for the firfl 6 days; the whole was 27. Ss. ; what was it per day?-Anf. 15. per day the first 3 days, 3s. per day the next 3 days, and 125. per day the 3 laft days.

Double pofition, or the double rule of falfe, is when two falfe pofitions are requifite to give an answer to the questioni. Rule 1. Take any two convenient numbers, and work with each of them according to the queftion, as in fingle pofition. 2. Find the difference between the refult of each of these falfe pofitions and the refult of the queftion; thefe differences are called the errors. 3. Multiply each error by the contrary position, that is, the first error by the second pofition, and the fecond error by the firft pofition; then find the fum and difference of the products. 4. If the errors are both alike, that is, if the refult of the two positions be both greater or both lefs than the refult of the question, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. But if the errors be unlike, that is, if the result of one pofition be greater and the other lefs than the truth, then the fum of the products must be divided by the fum of the errors, and the quotient will be the answer.

Example 1. Three perfans, A, B, C, built a houfe, which coft 50cl. of which B paid half as mnch again as A, and C paid as much as A and B together; what did each pay?

Difference of errors 3,0)300,0 difference of the products. £100 for A's fhare, wherefore

B must have paid 150 and D must have paid 250 which added togethergives 500

being half as much again as A, being as much as both A and B, the original fum.

From this example may be feen the method of working this rule, which is always the fame, except when the errors are unlike; then the fum of the products is to be the dividend, and the fum of the errors the divifor as above directed.

Qu. 2. A falefman bought a number of oxen, fheep, and lambs, for which he paid 1157.; for the oxen he paid 10l. each, for the sheep 20s. each, and for the lambs 10s. each; how many of each fort did he buy?-Anf. 10 of each. Qu. 3. Three perfons, A, B, C, have equal incomes; A faves of his income every year; B fpends 10l. per annum more than A, and C fpends 10l. per annum more than B. At the expiration of five years, C finds himself in debt 50%. what is each perfon's income, and what has each faved or fpent?-Anf. the income of each is tool. per annum; A has faved gol.; B has faved nothing; and C has spent 50%. more than his income.

Qu. 4. A labourer was hired for 40 days: for every day he wrought he was to receive 25. and for every day he was idle he was to forfeit 15.; at the end of the time he had to receive 445.; how many days did he work, and how many was he idle?-Anf. he wrought 23 days and was idle 12 days.

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