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Qu. 2. A owes Brocol. 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 77} months.

In this example, of roool. multiplied by 6 months produces 2, ani multiplied by 8, produces 3}, and is (the remainder of the 1000l.) muitiplied by 10, produces 2!? ; and these three products added together gives 7 months for the answer.

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

Qu, 3. A tradesman cwes his creditor 144/1; 447. 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 tradesman defiring to have more time for the payment of the last 40l. pays his creditor the 601. due 6 months after, in ready money ; how long may he defer paying this last 40l. to make bim amends for this prompt payment? - Anf. 17 months.

In this example, the 441. to be paid in ready money neglected; but for the 60!. paid 6 months before due, 'I find by the rule of three what interest it would gain at any rate per cent. in that time, and then how long the 401. may be lent for that interest at the same 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 u'ed by men of business, is not arathematically exact. The reason of the rule given by many writers, is, that for the debtor paying a sum before the time it is due, an equal 'um Mould be forborn, for as long a time after it is dae; but this is a mustake, for by the debtor paying money before it is due, he has the discount 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 such questions as cannot be resolved by any direct rule iu vulgar arithmetic, and must be performed by false or feigned numbers.

Position is either single or double.

Single position is when the question can be resolved by one false position or set of feigned numbers, and one operation in the rule of three direct.

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

Then lay (by the rule of three) as the result of this false operation is to any of its parts, so is the true result in the question to the corresponding part required.

Esample 1. A son asking his father his age, the father answered, I am double the age of your eldest brother John, and he is three times the age of your youngest brother Henry, and the sum of all our ages is 80 years; what is each person's age?

Falfe Suprofition. Suppose Henry's age is 6 Then John's age will be 18 And the father's age 36

The sum of the ages oo

The Truth.
Henry's age

8 John's

24 The father's 48

The sum 80

Here I suppose Henry's age to be 6, at which suppo Gtion John's age must be 18, the father's 36, and the sum of these three is 60, whereas it should be 80; then I say, as 6o the sum of the false supposition, is to do the sum of the true one, G

so

so is 6 the supposed 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 asked B how much money he had in his pocket; B answered, if you give me 4 guineas of the money in your pocket, I Mall have five times as much as you will then have ; but if, instead of that, I fiould give you a guineas of the money in my pocket, you will then have twice as much as I shall theu have: how much money had each ? - Ans. 6 guineas.

Qu. 3. A perfon hired a horse for 9 days, on the following terms: for the first 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 firft 6 days ; the whole was al. 8s. ; 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 last days.

Double pofition, or the double rule of false, is when two false postiops are requifite to give an answer to the questiori.

Rule 1. Take any two convenient nunibers, and work with each of them according to the question, as in fiagle position. 2. Find the difference between the result of each of these false positions and the result of the question ; these differences are called the errors. 3. Multiply each error by the contrary position, that is, the first error by the second position, and the second error by the first pofition; then find the sum and difference of the products. 4. If the errors are both alike, that is, if the result of the two positions be both greater or both less than the result 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 erros be unlike, that is, if the result of one position be greater and the other less than the truth, then the suin of the products must be divided by the sum of the errors, and the quotient will be the answer.

Example 1. Three persons, A, B, C, built a house, which cost socl. of which В paid half as mnch again as A, and C paid as much as A and B together; what did each pay?

Result 450

First Suprofition.

Second Suppofition. Suppole A paid £90 Suppofe A paid 696 Then B must have paid 135

Then B must have paid 144 And C must have paid 225

And C bave paid 240

Result 480 First erro 50

Second error 20 Second position 96

First position

90 300 Second product 1800

450
First product 4800

Second product 1800
Difference of errors 3,0)300,0 difference of the products.

£100 for A's Mare, wherefore B must bave paid 150 being half as much again as A, and D must have paid 250 being as much as both A and B which added togethergives £500 the original sum.

From this example may be seen the method of working this rule, which is always the same, except when the errors are unlike; then the sum of the products is to be the dividend, and the sum of the errors the divisor as above directer.

Qu. 2. A salesman bought a number of oxen, sheep, and Jambs, for which he paid 115l.; for the oxen he paid rol. each, for the sheep 20s. each, and for the lambs 1os. each; how many of each sort did he buy? --- Ans. 10 of each.

24. 3. Three persons, A, B, C, have equal incomes; A saves to of his income every year; B (pends rol. per annum more than A, and C spends rol. per. annum more than B. At the expiration of five years, C finds himself in debt zo. what is each person's income, and what has each saved or fpent? - Ans. the income of each is rool. per annum; A has saved gol.; B has saved nothing; and C has spent gol. more than his income.

Qu. 4. A labourer was hired for 40 days: for every day he wrought he was to receive as. and for every day he was idle he was to forfeit is.; at the end of the time he had to receive 445. ; how many days did he work, and how many was he idle ? - Ans, he wrought 28 days and was idle 12 days.

SECT,

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EXCHANGE is that rule which teachelh to find what sumof the money of one country is equal to a given sum of the money of another country, the course of exchange being known.

The course of exchange is that sum of the money of one country which is proposed to be given for a certain contant sum of that of another country: thus, when we say the course of exchange between England and Holland is 34). Flemish per pound sterling, 'it signifies that 1 pound sterling is equal to the value of

345: in Flemish money. This course of exchange varies on the part of the foreign coins, according to the state of public affairs. The

par of exchange is that quantity of the coin of one country which is intrinsically equal to a certain quantity of the coin of another country, according to the value of the metal.

Móst foreign countries have two sorts of coins, called current money and banco money ; the first is that in general use throughout the country; the latter is that kept in the banks of those places, and is finer than the other ; the difference between any suin as it is valued in current money and banco money is called the agio.

The money used in exchange is generally imaginary, and different from that in which the accompts are kept in most places: the money used in exchange also differs from current money in its value.

Before

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