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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. ; 1;lb. at 60c.
PROOF. ilb. at 18c., yain 22c. = 38c.,
ilb. at 18c. 18c. ilb. at 24c., gain 16c.
gain. ilb. at 24c. 24c. ilb. at 48c., loss 8c.
ilb. at 48c.
48c. ilb. at 60c., loss 20c. 38c.,
illb. at 60c. 90c. 4žlb.
180c. flb. at 60c., loss 10c.
180c. • 41 = 40c. per
lb. 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 ilh. 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 illb. 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 2
20 10 the rates of the articles, with the 217 16
8 4 given mean on the left, we con
16 8 nect together terms denoting the
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
lb. to gain 1ct.; and by taking ilb. at 60cts. the loss will be 20cts.; hence it will require zilb. to lose lct. Therefore, the gain on lb. at 18cts. balances the loss on lb. at 60cts. The proportions at these rates are, then, zand 2 or (by reducing to a common denominator) ter and to, 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 ráte. 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 lb. at 18cts., 1lb. at 24cts., flb. at 48cts., and zolb at 60cts.; or, 20lb. at 18cts., 8lb. at 24cts., 16lb. at 48cts., and 22lb. 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 vhich 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, 1gal. of 60 cents, 9gal. of 80 cents, and lgal. 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. OPERATION.
By taking loz. at 15 carats loz. at 15, gain 5) fine there is a gain of 5 carats, 20
loz. at 17, gain 3 =18 by taking loz. at 17 carats a 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 takBoz. 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 giren, 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 10} cents, and 6 pounds at 131 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 ?
511. When the quantity and rate of a mixture, with the rates of the articles composing it, are given, to find the quantity of each article which is not limited.
Ex. 1. How many gallons of water must be mixed with wine at $1.50 a gallon so as to make a mixture of 100 gallons worth $ 1.20 a gallon ?
Ans. 20 gallons.
Representing the S Igal. at 0.00, gain 1.20 1.20 rate of the water by gal. at 1.50, loss.30)
0.00, we then find, 1.20
as in Art. 509, the 3gal. at 1.50, loss.90
quantity required of
each article, in com1+4= 5gal.; } of 100gal. = 20gal. Ans. posing a mixture of
the given mean, to be 1 gallon of water and 4 gallons of the wine. Therefore, the quantity of water is to the whole quantity of the mixture as 1 to 5. Hence, in a mixture of 100 gallons at the mean rate given, the water must be } of 100 gallons, or 20 gallons.
RULE. — Find the proportional quantities of the several articles, as in Art. 509, or 510, as though the quantity of the mixture were not limited.
Then take such a part of the given quantity of the mixture, as each of these proportional quantities is of their sum.
2. A merchant has sugar at 8 cents, 10 cents, 12 cents, and 20 cents a pound; with these he would fill a hogshead that would contain 200 pounds. How much of each kind must he take, so that the mixture may be worth 15 cents a pound?
Ans. 33žlb. of 8, 10, and 12cts., and 100lb. of 20ets. 3. How much wheat at $ 2.00 and $ 1.80 a bushel must be mixed with 4 bushels at $ 2.20, and 10 bushels at $ 1.70, so as to make a mixture of 48 bushels, worth $ 1.90 per bushel ?
Ans. 21 bushels $ 2.00; 13 bushels at $ 1.80. 4. How much gold of 15, 17, and 22 carats fine must be mixed with five ounces of 18 carats fine, to make a composition of 5 pounds, that shall be 20 carats fine ?
5. A gentleman's servant having been ordered to purchase 20 animals for $ 20, brought home sheep at $ 4.00, lambs at $ 0.50, and kids at $ 0.25 each. Required the number of each kind.
Ans. 3 sheep ; 15 lambs ; and 2 kids.
1. A manufacturer employs a number of men at $ 1.20, and a number of boys at $ 0.80, per day; and the amount of the wages of the whole is the same as if each had $ 0.974 per day. Required the number of men, that of the boys being 9.
Ans. 7 men. 2. What is the value of 5000 specie rix dollars 12 skillings of Sweden in United States money ? Ans. $ 5300.265.
3. Exchange between New Orleans and England being in New Orleans at 8 per cent. premium, and in Liverpool at 10 per cent. premium, if L. Sandford of Liverpool owes M. Lassale of New Orleans for cotton to the amount of 1500£. 15s. sterling, what will be the difference between Lassale drawing or Sandford remitting the amount ?
4. If 17 gallons of spirits at $ 1.26 per gallon be mixed with 7 gallons at a different price, and 20 per cent. be made by selling the mixture at $ 1.56, what was the price of the latter kind per gallon?
Ans. $ 1.39% per gallon. 5. What is the value of 100 ounces 20 tari 10 grani of Sicily in lire and centesimi of Leghorn?
Ans. 1510 lire 25 centesimi. 6.0 If 20 United States gallons equal 1 eimer of Sweden, 3 eimers of Sweden equal 4 eimers of Trieste, 24 eimers of Trieste equal 9 ahms Danish, and 33 ahms Danish equal 5 carri of Naples, which will cost the most in United States money, 170 eimers of Trieste of wine at 1 florin 45 kreutzers per gallon, or 12 carri of wine at 1 ducat of Naples a gallon?
7. When exchange on England is at 8 per cent. premium, and freight at 12d. per United States bushel, how much can be paid per bushel for wheat in Baltimore, in answering an order from Liverpool limited to 60s. per imperial quarter?
Ans. $ 1.50 61 per bushel. 8. A merchant mixes 11 pounds of tea with 5 pounds of an inferior quality, and gains 16 per cent. by selling the mixture at 87 cents per pound. Allowing that a pound of the one cost 12 cents more than a pound of the other, what was the cost of each kind per pound ?
Ans. The one 78 cts. ; the other 66cts. per lb.