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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 7019 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 91 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 į 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, 165 per cent.
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.
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 2cwt. 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 $18
Since 2cwt. at $ 9 per cwt. is worth 7 x 1
7 $ 18, 1cwt. at $ 7 per cwt. is worth $ 7, 10 x 2
and 2cwt. at $ 10 per cwt. is worth $ 20;
2cwt. + lcwt. + 2cwt. 5cwt. is worth 5) $ 45 $ 18 + $ 7 + $ 20 = $ 45; and icwt. is
worth as dollars as 45 contains times $ 9 Ans.
5, or $ 9. 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. $ 0.473.
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. llb. at 18c.; lib. at 24c.; llb. at 48c.; 14lb. at 60c.
ilb. at 18c., yain 22c. = 38c.,
ilb. at 18c. 18c. ilb. at 24c., gain 16c. S gain. ilb. at 24c. = 24c. ilb, at 48c., loss 8c.
ilb. at 48c. : 48c. ilb. at 60c., loss 20c.
180c. 31b. at 60c., loss 10c.
180c. • 41= 40c. per Compared with the given mean value, by taking ilb. at 18cts. there is a gain of 22cts., by taking ilb. at 24cts. a gain of 16cts., by taking ilb. at 48cts. a loss of 8cts., and by taking llb. 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 lb. more of the spice at 60cts.; and thus have a mixture of the required average value, by having taken, in all, ilb. at 18cts., ilb. at 24cts., llb. at 48cts., and i£lb. at 60cts. We prove the correctness of this result by dividing the value of the whole mixture by the number of pounds taken.
Having arranged in a column 18 22 20 10
the rates of the articles, with the 24
8 4 10
given mean on the left, we con
16 8 nect together terms denoting the 60. 26 22 11
rate of the articles, 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 ilb. at 18cts. the gain will be 22cts.; hence it will require
Alb. to gain 1ct.; and by taking ilb. at 60cts. the loss will be 20cts.; hence it will require molb. to lose 1ct. Therefore, the gain on zblb. at 18cts. balances the loss on zölb. at 60cts. The proportions at these rates are, then, 22 and zo or (by reducing to a common denominator) 4 and 4, 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 zb. at 18cts., 1 lb. at 24cts., flb. at 48cts., and Zolb. at 60cts.; or, 20lb. at 18cts., 8lb. at 24cts., 16lb. 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 given 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 60 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. Igal. of water, lgal. of 60 cents, 9gal. of 80 cents, and Igal. of $ 1.20.
510. When the quantity of one or more of the articles composing a mixture of a given mean value is given, to find the quantity of each of the others.
Ex. 1. How much gold of 15, 17, and 22 carats fine must be mixed with 5 ounces of 18 carats fine, so that the composition may be 20 carats fine ? Ans. loz. at 15 carats, loz, at 17 carats, 9oz. at 22 carats.
By taking loz. at 15 carats 1oz. at 15, gain 5
fine there is a gain of 5 carats, loz. at 17, gain 3 18 by taking loz. at 17 carats a 20 5oz. at 18, gain 10
gain of 3 carats, by taking 5oz., loz. at 22, loss 2
the given quantity, at 18 carats,
= 18 a gain of 10 carats, and by tak8oz. at 22, loss 16
ing loz. at 22 carats a loss of 2
carats; and to balance the gain and loss we take 8oz. additional, at 22 carats, a loss of 16. We have then for the result loz. at 15 carats, loz, at 17 carats, and 9oz. at 22 carats.
RULE. - Take of the limited article or articles the quantity or quantities given, with a unit of each of the other articles of the proposed mixture, and note the gain or loss; and then take, if required, such additional quantity or quantities of the articles not limited, as shall equalize the gain and loss.
per bushel ?
2. How much wine at $1.75 and at $ 1.25 per gallon must be mixed with 20 gallons of water, that the whole may be sold at $ 1.00 per gallon ?
Ans. 20 gallons of each. 3. How much wheat at $ 2.00 per bushel and at $ 1.80 per bushel must be mixed with 4 bushels at $ 2.20 per bushel and 10 bushels at $1.70 per bushel to make a mixture worth $1.90
Ans. 9 bushels at $ 2.00; 1 bushel at $ 1.80. 4. How many pounds of sugar, at 8, 14, and 13 cents a pound, must be mixed with three pounds at 9 cents, 4 pounds at 102 cents, and 6 pounds at 13.cents a pound, so that the mixture may be worth 124 cents a pound ?
Ans. llb. at 8cts.; 5 lb. at 13cts.; and 9lb. at 14cts. 5. How much barley at 45 cents a bushel must be mixed with 10 bushels of oats at 58 cents a bushel, to make a mixture worth 50 cents a bushel ?