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in the ratio of to, or 1 to 3; for, in this instance, there is as much lost on 1 gallon at 6 dollars, as is gained on 3 gallons at 2 dollars.

We see by the preceding ratios, that the nearer the value of the mixture is to that of one of the ingredients, the greater must be the relative quantity of this ingredi ent, in forming the compound; and the farther the value of the mixture is from that of one of the ingredients, the less must be the relative quantity of this ingredient in making the compound.

Hence, if we make the difference between the rate of each ingredient and that of the compound, the denominator of a fraction having 1 for its numerator, these fractions express the ratio of the ingredients required to make the compound; and, when these fractions are reduced to a common denominator, the numerators express the required ratio of the ingredients.

The

If, for example, it be required to mix gold of 12 carats fine with gold of 22 carats fine, in such proportion that the mixture may be 18 carats fine, we can ascertain the proportion of each kind in the following manner. difference between 18 and 12 is 6; making 6 the de nominator of a fraction with 1 for its numerator, we have the fraction; taking the difference between 18 and 22, we in like manner obtain the fraction; therefore, the fractions, and, express the required proportion of each sort of gold. These fractions, when reduced to a common denominator, re 24 and 24, and the numerators express the required proportion of each sort. Therefore, we must take 4 grains of 12 carats fine, and 6 grains of 22 carats fine; or, in that ratio.

If, for a second example, we would make a mixture 18 carats fine from gold of 15 carats and 20 carats fine, we should, in the same manner, obtain the fractions, 2 and %, to express the required proportion of the two sorts of gold; consequently, in this instance, we should take 2 grains of 15 carats fine, and 3 grains of 20 carats fine. Therefore, since the fineness of the compound is the same in both the preceding examples, if we would make a compound 18 carats fine, from the four kinds of gold

mentioned in the two examples, we should take 4 grams of 12 carats, 6 grains of 22 carats, 2 grains of 15 carats, and 3 grains of 20 carats fine.

Now, these results may be readily obtained by writing the rates of the given simples one under another, in regular order, beginning either with the least or greatest, and alligating one of a less with one of a greater rate than that of the compound, and writing the difference between the rate of each simple and the rate of the compound, against the rate of the simple with which it is alligated. Thus, 12. 4 grains 12 carats fine

15

2 66

15 66 66

18

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We may connect the rates of the simples differently and obtain equally correct, but different results.

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It mu

be observed, that the two simples linked together, must always be one of a less, and the other of a greater rate, than the rate of the compound.

By connecting a less rate with a greater, and placing the differences between them and the mixture rate alternately, the gain on the one is precisely balanced by the loss on the other. This being true of every two, it is true of all the simples in the question, whatever may be their number.

It is obvious, that a question in Alligation Alter. admits of a great variety of answers, all agreeing with the requisition of the question; for we may variously alligate the values of the ingredients, and thus obtain various results, all of which will be correct; and we may add all these together, and the results will be correct answers. We may also multiply, or divide the quantities found; for, if two quantities of two simples make a balance of loss and gain in relation to the value of the compound, so must also the double or treble, the half or third part, or any other ratic of the quantities.

We shall give the questions in Alligation Alternate under four cases.

CASE I. The ratios of the several ingredients being given, to make a compound of a fixed rate.

RULE. First-Write the rates of the several ingredients in a column under one another.

2dly-Connect with a continued line the rate of each ingredient less than the rate of the compound, with one or more rates greater than the rate of the compound; and each of a greater rate than the rate of the compound with one or more of a less rate.

3dly-Write the difference between the rate of each ingredient and the rate of the compound, opposite the rate of the ingredient with which it is connected.

4thly-If only one difference stand against any rate, it will be the required quantity of the ingredient of that rate; but, if there be several, their sum will be the quantity required.

7. A goldsmith has gold of 17, 18, and 22 carats fine, and also pure gold. What proportion of each sort musi he take, to compose a mixture 21 carats fine?

8. Having gold of 12, 16, 17, and 22 carats fine, what proportion of each kind must I take, to make a compound 18 carats fine?

9. A merchant has spices at 30, 33, 67, and 86 cents a pound. How much of each sort must he take, to make a mixture worth 56 cents a pound?

10. A wine merchant nas Canary wine at 50 cents a gallon, Sherry at 76 cents, and Claret at 175 cents per gallon. How much of each sort must he take, to make a mixture worth 87 cents a gallon?

11. A goldsmith wishes to mix gold of 16, 18, 19, and 23 carats fine, with pure gold, in such proportions that the composition may be 20 carats fine. What quantity of each must he take?

12. It is required to mix different sorts of wine, at 56, 62, and 75 cents per gallon, with water, in such propor tions that the mixture may be worth 60 cents a gallon. How much of each must be taken?

13. How much corn at 52 cents a bushel, rye at 56 cents, wheat at 90 cents, and wheat at 1 dollar a bushel, must be mixed together, that the composition may be worth 62 cents a bushel?

14. A silversmith wishes to mix alloy with silver of 10, and 7 ounces fine, and pure silver, in such proportion that the mass may be 9 ounces fine: 12 ozs. fine being pure. How much of each must he take?

CASE II. When one of the ingredients is limited to a certain quantity.

RULE. Find the quantity of each ingredient, as in Case 1st. in the same manner, as though no such limitation were made; then as the difference against that simple, whose quantity is given, is to each of the other differences, so is the given quantity of that simple to the quantity required of each of the other siniples.

15. A trader has 90 pounds of tea worth 40 cents a pound, which he would mix with some at 50 cents, some at 85 cents, and some at 90 cents. How much of each of the other sorts must he mix with the 90 pounds, to make a mixture worth 60 cents a pound?

First solution.

Second solution.

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30 1090 : 30

30: 2090 : 60

Ans. 75 lb. at 50 cents, 30lb. at 85 cents, and 60 pounds at 90 cents.

thus 25 30=90: 108

25: 20 90:

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72

36

Ans. 108 lb. at 50 cents, 72lb. at 85 cents, and 36 pounds at 90 cents.

16. A farmer wishes to mix corn at 54 cents a bushel, rye at 61 cents a bushel, and wheat at 96 cents a bushel, with 3 bushels of wheat worth 1 dollar and 10 cents a bushe.. How much of each of the other three must be mixed with the 3 bushels of wheat at 1 dollar and 10 cents a bushel, that the mixture may be worth 75 cents a bushel ?

17. How much gold of 16, 20, and 24 carats fine, and how much alloy, must be mixed with 10 ounces of 18 carats fine, that the composition may be 22 carats fine?

18. How much silver of 6.5 ounces fine, and of 10.5 ounces fine, and alloy, must be mixed with 17.1 ounces of pure silver, that the mass may be 9.5 oz. fine?

It must be observed, that pure silver is 12 ounces fine.

CASE III.

When two or more of the ingredients are

limited in quantity.

RULE. Find, as in Alligation Medial, what will be the rate of a mixture made of the given quantities of the limited ingredients only; then consider this as the rate of a limited ingredient, whose quantity is the sum of the quantities of the limited ingredients, from which, and the rates of the unlimited ingredients, proceed to calculate the several quantities required, as in Case 11.

19. I have 18 gallons of wine at 48 cents a gallon, 8 gallons at 52 cents, and 4 gallons at 85 cents, and would mix the whole with two other kinds of wine, one at a dollar and 26 cents, the other at 2 dollars and 12 cents a gallon. How much of the wine at a dollar and 26 cents, and of that at 2 dollars and 12 cents, must I mix with the other three, that the mixture may be worth a dollar a gallon?

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The 30 gal. come to

$16.20, which is .54 a gallon. 54 cents a gallon being the mean value of the 30 gallons, contained in the three kinds that are limited, I must now inquire how much of each of the other two sorts of wine at 1 dollar 26 cents, and 2 dollars 12 cents, must be mixed with 30 gallons at 54 cents a gallon, to make a mixture worth one dollar a gallon.

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