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CHAP. XI

ALLIGATION.

78 that Rule whereby we resolve Questions concerniag the mixing of several Simples, of commodities into one Compound Quantity.

Alligation is either Medial or Alternate.

Alligation Medial, is when having the several quantities and rates of divers simples proposed, we discover the rate of a mixture competinded of these simples:

Rule

Find according to the given rates, the value of each given quantity, then taking the sum of these quantities; and the sum of their values, say, if that suiñì of quant * ties give that sum of values what will the quantity (pro posed) give?

Applications

1. Suppose 13 bushels of wheat, at 5, the bushel, and 12 bushels of rye, at 3s. 6d. bushel, were mixed toges ther, what is the mean rate or price it may be sold at a bushel, without loss or gain.

s. d.

1.

15 Bushels, at 5

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The reason of this operation is manifest in itself.

To prove the truth of the Work.

Find the value of all the mixture at the mean rale ! Likewise the value of each particular quantity proposed to lie mixed at its given rates, and collect these particular values into one sum, if the said sun is equal to the value of All the mixture before found, the work is right, as

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2. A tobacconist mixeth 361. of tobacco, worth 1s. 6d. a pound, with 12. of another sort, at 2s. a pound; and 12. of a third sort, at Is. 10d. How may he sell the mixture è lb. ? Answ. 1s. 8d.

3. A vintner mixeth 311⁄2 gallons of Malaga sack, worth 7s. 6d. the gallon, with 18 gallons of Canary, at 63. 9d. the gallon; 13 gallons of sherry, at 5s. the gallon; and 27 gallons of white wine, at 4s. 3d. the gallon. 'Tis required to find what one gallon of this mixture is worth?

Answ. 6s. gallon.

4. A druggist has 200. of ginger at 6d. the b. 30015. at 7d. the . and 400. at Sd. the b. Now if he mixes them together, what will 1 cost? Answ. 7 d.

5. 100 casks of butter, qt. 200Cwt. 3qrs. 14b. at 18s. 8d. C. are mixed with 150 casks, qt. 306C. 1qr. 7th. at 23s. 4d. C. I demand what 1C, of said butter stands him? Answ. 23s. 10d.

6. There are melted and mixed together two sorts of silver; one sort is worth 5s. and the other 4s. an ounce; and there were 4 ounces of the first, and 8 ounces of the latter what is the value of one ounce of this mixture?

Answ. 4s. 4d.

7. A goldsmith melts 8. 5 oz. of gold bullion of 14. caracts fine, with 12 84oz. of 18 caracts fine. I demand how many caracts fine this mixture is?

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8. A refiner has 10. of gold of 20 caracts fine, and melts it with 16. of 18 caracts fine. The question is, how much alloy must be put to it to make it 22 caracts fine? Answ.

Аа

* An ounce of pure gold being reduced into 24 equal parts, these parts are called caracts: but gold is often mixed with some baser metal, which in the mixture is called the alloy and according to the propor tion of pure gold which is in every ounce, so the mixture is said to be so many caracts fine: Thus, if only 22 caracts of pure gold, and two of alloy, it is 22 caracts fine; if 20 caracts of pure gold and 4 of alloy it is 20 caracts fine; if there is no alloy, it is 24 caracts fine, or pure gold.

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Answ. 'Tis not fine enough by 3 caracts, so no alloy must be put to it but more gold.

A

SECT II.

ALLIGATION ALTERNATE.

LLIGATION Alternate is, when we have the several ingredients to be mixed, and the mean rate of the mixture given, to find such quantities of the simples or ingredients, as being mixed together, shall bear that common

rate.

Rule.

The rates being all of, or reduced to one denomination, 2. Set the rates of the simples in a column under one another, and the mean rate on the left-hand of these. 3. Connect or link together the several simple rates, so that every one less than the mean be linked to one or more greater than it, and every one greater with one or more less. 4. Take the difference between the mean rate and that of the several simples, and write it over against all the simples with which that one (whose difference it is) is linked, then the sums of these differences standing against every simple rate, are such quantities of the several simples against which they stand as answer the question.

I. When the simple rates do not exceed three, there can be but one way of linking them; because the mixture or mean rate must be between the highest and lowest rate of the simples; else, 'tis plain the mixture would not bear that rate, but would be of a greater or lesser rate, as the simples were either all of a greater or lesser: So then of two simples one will be greater and one less than the mean rate, and of three one greater and two less, or two greater and one less, which cases can admit but one way of linking.

Examples.

1. How much wheat at 63. the bushel, and barley at 35. 8d. bushel, will make a mixture that may stand in 1s. 4d. the bushel ?

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The Proof.

Questions in this Rule are proved by Alligation Medial, as follows.

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2. How much tobacco of 74d. the b. and 93d. the fb. must be mixt so that it will stand in 84d, the b Answ. 14. of 74d. and I. of 93d.

3. What quantity of sugar at 11d. the . and 74d. the b. would make a mixture so that it would stand in 104d. the fb.? Answ. 3. at 11d: and b, at 74d.

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4. How many oz, of silver; of 11 oz. fine, and 8 oz. fine, must be melted together to make the mass or mixture 9 oz. fine? Answ. 1oz. of 11oz. fine, and 2 oz. of 8 oz, fine.

II. When one rate is joined to two others, the sum of the differences of the said two and the mean rate, will be the quantity sought, at that rate to which the two are linked.

5. A merchant hath sugar of 5d. 10d. and 12d. t. How much of each sort must he take that he may sell a pound for 8d.

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Answ. 6 bushels of rye, 6 bushels of barley, and 24

bushels of oats.

A a 2

7. A

7. A grocer would mix three sorts of sugar together, viz. one sort at 10d. b. another at 7d. and another at 6d. How much of each sort must he take, that the whole mixture may be sold for Ed.

Answ.

tb. d.

b.

3 at 10 4th.

2 at 7

2 at 6

8. A vintner has brandy, at 3, 5, and 6s. the gallon, and has a mind to mix a quantity of them together, so that it may stand him in 5s. 6d. the gallon. I demand how many gallons he must take of each sort?

Answ. at 3s. at 5s and 3 at 63.

III. If the number of simple rates exceed 3 there may be several ways of linking them, and every way brings different answers; but all giving such numbers as will auswer the thing required,

Examples.

9. A merchant would mix wines, at 14, 15, 19 and 22s. dozen, so as the mixture may be worth 18s. What quantity of each may be taken?

This sum may be linked 7 different ways as follow, viz.

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Besides the different answers produced by this different manner of linking the terms or simple prices, questions in Alligation Alternate (being of that kind Algebraists term unlimited problems) have an infinite variety of other answers for any other numbers in proportion to those found by this Rule (as above) will answer the questions as well

as those.

10. A

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