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504." To compute arbitration of exchanges.

Ex. 1. When the exchange between New York and London is at a premium of 9 per cent., and that between London and Paris 25 francs to a pound sterling, how much must be paid in New York for a bill on Paris for 1000 francs?

Operatic!.. Since $44 =

$40 — 9£. l£. of the nomi

109£. = 1 00£. nal par of ex

1 £. = 2 5 fr. change (Article

lOOOfr. = $— 497), $40=9£.;

IQ j0 and 109£.of the

vw „„ same value = 40 X 109 X 1 XX000 = + Ans 100£. at 9 per

9 X X00 X 20 cent . PreTM"111

AVe write the terms of equivalent value as antecedent and consequent, and proceed as in conjoined proportion (Art. 341).

Note. — When it is required to find which of several routes of exchange is the most advantageous, the rate of exchange by each route may be determined first, and the results then compared.

Examples.

2. When exchange at Lisbon on Paris is at the rate of 5 francs 95 centimes per millrea, and at Paris on the United States at 5 francs 20 centimes per dollar, how much must be paid in Lisbon to cancel a demand in New York for $ 3500?

Ans. 3058 millreas 8"23T9y reas.

3. A merchant in Boston wishes to pay 2000£. in Liverpool. Exchange on Liverpool he finds is at 10 per cent, premium, on Paris 5 francs 20 centimes to a dollar, and on Hamburg 35 cents to a mark banco; and the exchange between France and England at the same time is 24 francs to a pound sterling, and that of Hamburg on England 13§ marks banco to a pound sterling. Which is the most advantageous course of remittance, that direct to Liverpool, or that through Paris or Hamburg?

4. A merchant of St. Louis wishes to pay a debt of $ 5000 in New York. The direct exchange is 1£ per cent, in favor of New York, but on New Orleans it is £ per cent. discount; and between New Orleans and New York at £ per cent, premium. How much would he save by the circular exchange compared with the direct? Ans. $87.56^.

5. A merchant in Boston owes a debt of 9760 thalers in Bremen, to pay which he purchases a bill on London, at a premium of 9 per cent., and remits the same to his agent in England, on whom his creditor is requested to draw. If the exchange between London and Bremen be at the rate of 34d. sterling per thaler, and the charges for brokerage £ per cent., how much must have been the cost of the bill in New York? Ans. $ 6731.74+.

6. When exchange between New Orleans and Hamburg is at 34 cents per mark banco, and between Hamburg and St. Petersburg is 2 marks 8 schillings per ruble, how much must be paid in St. Petersburg for a bill on New Orleans for $ 650? Ans. 764 rubles 70}f kopecks.

7. When exchange in Philadelphia on Boston is at £ per cent, premium, and on Chicago at 2 per cent, discount, if the exchange between Chicago and Boston is at par, how much better is the circuitous route of exchange between Philadelphia and Boston than the direct?

8. A merchant, about to import broadcloth, finds he can obtain the quality desired in Amsterdam at 8 guilders per Amsterdam ell; in Berlin, at 3 thalers 15 groschen per Berlin ell; and in England, at 15 shillings per yard. Exchange being on Amsterdam at 40 cents per guilder, on Berlin at 66 cents per thaler, and on England at 9^- per cent, premium, and the freight being the same in each case, from which place can he make the importation to the best advantage? Ans. Berlin.

9. When exchange between Washington and London is at 8 per cent, premium, and between London and Paris 25.25 francs per pound sterling, what sum in Washington is equal to 7000 francs in Paris?

10. A merchant in London remits to Amsterdam 1000£. at the rate of 18d. per guilder, directing his correspondent at Amsterdam to remit the same to Paris at 2 francs 10 centimes per guilder, less J per cent, for his commission; but the exchange between Amsterdam and Paris happened to be, at the time the order was received, at 2 francs 20 centimes per guilder. The merchant at London, not apprised of this, drew upon Paris at 25 francs per pound sterling. Did he gain or lose, and how much per cent.? Ans. Gain, 16ff per cent,

ALLIGATION.

505. Alligation is a process employed in the solution of questions relating to the compounding or mixing of articles of different qualities or values.

It is of two kinds: Alligation Medial, and Alligation Alternate.

ALLIGATION MEDIAL.

506. Alligation Medial is the process of finding the mean or average rate of a mixture composed of articles of different qualities or values, the quantity and rate of each being given.

507. To find the average value of several articles mixed, the quantity and rate of each being given.

Ex. 1. A grocer mixed 2ewt. of sugar worth $9 per cwt. with lcwt. worth $7 per cwt. and 2cwt. worth $ 10 per cwt.; what is lcwt. of the mixture worth? Ans. $ 9.

$9 X 2 = $ 1 8 Since 2cwt. at $9 per cwt. is worth

7X1= 7 $ 18, lcwt- at $ 7 per cwt. is worth $ 7,

l 0 ^ 2 = 2 0 anl^ 2cwt- at $ 10 per cwt. is worth $ 20;

— 2cwt. -f- lcwt. -f- 2ewt. = 5cwt. is worth

5) $45 $18+ $7 + $20 = $45; and lcwt. is

at n . worth as many dollars as 45 contains times $9 Ans. 5, or$g

Rule. -— Find the value of each of the articles, and divide the sum of their values by the number denoting the sum of the articles. The quotient will be the average value of the mixture.

Examples.

2. If 19 bushels of wheat at $ 1.00 per bushel should be mixed with 40 bushels of rye at $0.66 per bushel, and 11 bushels of barley at $ 0.50 per bushel, what would a bushel of the mixture be worth? Ans. $ 0.727|.

3. If 3 pounds of gold of 22 carats fine be mixed with 3 pounds of 20 carats fine, what is the fineness of the mixture? Ans. 21 carats.

4. If I mix 20 pounds of tea at 70 cents per pound with 15 pounds at 60 cents per pound, and 80 pounds at 40 cents per pound, what is the value of 1 pound of this mixture?

Ans. $ 0.47^|.

ALLIGATION ALTERNATE.

508. Alligation Alternate is a process of finding in what ratio, one to another, articles of different rates of quality or value must be taken, to compose a mixture of a given mean or average rate of quality or value.

509. To find the proportional quantities of the articles of different rates of value that must be taken to compose a mixture of a given mean rate of value.

Ex. 1. A merchant has spices, some at 18 cents a pound, some at 24 cents, some at 48 cents, and some at 60 cents. How much of each sort must be taken that the mixture may be worth 40 cents a pound?

Ans. l1b. at 18c; lib. at 24c; l1b. at 48c; l£lb. at 60c.

FIRST OPERATION. PROOF.

(l1b. at 18c, gain 22c. > = 88c., l1b. at 18c. = 18c.

lib. at 24c, gain 16c. J 8""- l1b. at 24c. = 24c.

lib. at 48c, loss 8c. "] l1b. at 48c. = 48c.

lib. at 60c, loss 20c. [=38c., 1 J1b. at 60c. = 90c.

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Compared with the given mean value, by taking l1b. at 18cts. there ii a gain of 22cts., by taking l1b. at 24ets. a gain of 16cts., by taking lib. at 48cts. a loss of 8cts., and by taking l1b. at 60cts. a loss of 20cts. Now it is evident that the mixture, to be of the mean or average value given, should have the several items of gain and loss in the aggregate exactly offset one another. This balance we effect by taking J1b. more of the spice at 60cts.; and thus have a mixture of the required average value, by having taken, in all, l1b. at 18cts., l1b. at 24cts , l1b. at 48ets., and 141b. at 60cts. We prove the correctness of this result by dividing the value of the whole mixture by the number of pounds taken.

Second Operation. Having arranged in a column

18 I 20 10 the rates of the articles, with the

8 4 „ given mean on the left, we con£ 0r 16 or 8 ncct together terms denoting the

i 22 11 ra'e of tne artides, so that a rate

^ less than the given mean is united

with one that is greater. We then proceed to find what quantity of each of the two kinds whose rates have been connected can be taken, in making a mixture, so that what shall be gained on the one kind shall be balanced by the loss on the other.

By taking l1b. at 18cts. the gain will be 22ets.; hence it will require

40

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Jj1b. to gain let.; and by taking l1b. at 60ct*. the loss will be 20ets.; hence it will require ^y1b. to lose let. Therefore, the gain on ^j1b. at 18cts. balances the loss on ^1b. at 60cts. The proportions at these rates are, then, -fa and -g^ or (by reducing to a common denominator) ffis and ^jj, or (by omitting the denominators, which do not affect the ratio) 20 and 22, which is obviously the same result as would be obtained by placing against each rate the difference between the rate with which it is connected and the mean rate. In like manner we determine the quantity that may be taken of the other two articles, whose rates are connected together.

We thus find that there may be taken j^1b. at 18cts., ^1b. at 24cts., \\b. at 48cts., and ^5lb at 60cts.; or, 201b. at 18cts., 81b. at 24ets., 161b. at 48cts., and 221b. at 60cts. By dividing the last set by 2, we obtain another set of results, and by multiplying or dividing any of these results others may be found, all of which can be proved to satisfy the conditions of the question. Hence, examples of this kind admit of an indefinite number of answers.

Rule 1.— Take a unit of each article of the proposed mixture, and note the gain or loss; and then take such additional quantity or quantities of the articles as shall equalize the gain and loss. Or,

Rule 2. —• Write the rates of the articles in a column, with the mean rate on the left, and connect the rate of each article which is less than the gii en mean with one that is greater; the difference between the mean rate and that of each of the articles, written opposite to the rate with which it is connected, will denote the quantity to be taken of the article corresponding to that rate.

Note. — When a rate has more than one rate connected with it, the sum of the differences written against it will denote the quantity to be taken. There will be as many different answers as there are different ways of connecting the rates; and, by multiplying and dividing, these answers may be varied indefinitely.

Examples.

2. How much barley at 45 cents a bushel, rye at 75 cents, and wheat at $1.00, must be mixed, that the composition may be worth 80 cents a bushel?

Ans. 1 bushel of rye, 1 of barley, and 2 of wheat.

3. A goldsmith would mix gold of 19 carats fine with some of 15, 23, and 24 carats fine, that the compound may be 20 carats fine. What quantity of each must he take?

Ans. loz. of 15 carats, 2oz. of 19, loz. of 23, and loz. of 24.

4. It is required to mix several sorts of wine at 6O cents, 80 cents, and $1.20, with water, that the mixture may be worth 75 cents per gallon; how much of each sort must be taken?

Ans. lgal. of water, lgal. of 6O cents, 9gal. of 80 cents, and lgal. of $ 1.20.

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