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ALLIGATION ALTERNATE is the method of finding what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate. So that it is the reverse of Alligation Medial, and may be proved by it.
1. Set the rates of the simples in a column under each other.--2. Connect, or link with a continued line, the rate of each simple, which is less than that of the compound, with one, or any number, of those that are greater than the compound; and each
greater rate with one or any number of the less.--3. Write the difference between the mixture rate, and that of each of the simples, opposite the rate with which they ate linked.--4. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity.
The examples may be proved by the rule for Alligation Medial.
* Demonst. By connecting the less rate to the greater, and placing the difference between them and the rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss opon the whole is equal, and is exactly the proposed rate : and the same will be true of any other two simples managed according to the Rule.
In like manner, whatever the number of simples may be, and with how many soever every one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. Q. E. D.
It is obvious, from this Rule, that questions of this sort admit of a great variety of answers; for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found, by 2, or 3, or 4, &c: the reason of which is evident : for, if two quantities, of two simples, make a balance of loss and gain, with respect to the mean price, so must also the double or treble, the ļor part, or any other ratio of these quantities, and so on ad infinitum.
These kinds of questions are called by algebraists indeterminate or unlimited problems; and by an analytical process, theorems may be raised that will give all the possible answers.
1. A merchant would mix wines at 16s, at i8s, and at 22s per gallon, so as that the mixture may be worth 20s the gallon : what quantity of each must be taken?
16 2 at 165
2 at 18s
22 4 + 2 = 6 at 22s. Ans. 2 gallons at 16s, 2 gallons at 18s, and 5 at 225. 2. How much wine at 6s per gallon, and at 4s per gallon, must be inixed together, that the composition may be worth 5s per gallon?
Ans. 1 qt, or 1 gall, &c. 3. How much sugar at 4d, at 6d, and at 11d per lb, must. be mixed together, so that the composition formed by them may be worth id
lb ? Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity
of each sort. 4. How much corn at 2s 6d, 3s 8d, 4s, and 4s 8d per bushel, must be mixed together, that the compound may be worth &s 100 per bushel ?
Ans. 2 at 2s 6d, 2 at 3s 8d, 3 at 4s, and 3 at 4s 8d. 5. A goldsmith has gold of 16, of 18, of 23, and of 24 caracts fine : how much must he take of each, to make it 21 caracts fine ? Ans. 3 of 16, 2 of 18, 3 of 23, and 5 of 24.
6. It is required to mix brandy at 12s, wine at 10s, cyder at 1s, and water at O per gallon together, so that the mixture may be worth &s per gallon? Ans. 8 gals of brandy, 7 of wine, 2 of cyder, and 4 of water.
When the whole composition is limited to a certain quantity : Find an answer as before by linking; then say, as the sum of the quantities, or differences thus determined, is to the given quantity; so is 'each ingredient, found by linking, to the required quantity of each.
1. How much gold of 15, 17, 18, and 22 caracts fine, must be mixed together, to form a composition of 40 oz of 20 caracts fine ?
16 Then, as 16 : 40 :: 2 : 5
16 : 40 :: 10 : 25 Ans. 5 oz of 15, of 17, and of 18 caracts fine, and 25 oz of
22 caracts fine*.
Ex. 2. A vintner has wine at 45, at 5s, at 5s 6d, and at 6s a gallon; and he would make a mixture of 18 gallons, so that it might be afforded at 5s 4d per gallon; how much of each sort must he take ?
Ans. 3 gal. at 45, 3 at 5s, 6 at 5s 6d, and 6 at 6s.
* A great number of questions might be here given relating to the specific gravities of metals, &c. but one of the most curious may here suffice.
Hiero, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but suspecting the workman had debased it by mixing it with silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown.
Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of silver or copper, and each of the same weight with the former ; and by putting each separately into a vessel full of water, the quantity of water expelled by them determined their specific gravities; from which, and their given weights, the exact quantities of gold and alloy in the crowa may be determined.
Suppose the weight of each crown to be 10lb, and that the water expelled by the copper or silver was 92lb, by the gold 521b, and by the compound crown 64b; what will be the quantities of gold and alloy in the crown?
The rates of the simples are 92 and 52, and of the compound 64; therefore
12 of copper
28 of gold And the sum of these is 12 +28 = 40, which should have been but 10; therefore by the Rule,
40 : 10 :: 12 : 31b of copper }the answer.
When one of the ingredients is limited to a certain quantity; Take the difference between each price, and the mean rate as before, then say, As the difference of that simple, whose quantity is given, is to the rest of the differences se'verally; so is the quantity given, to the several quantities required.
1. How much wine at 5s, at 5s 6d, and 6s the gallon, must be mixed with 3 gallons at 4s per gallon, so that the mixture may be worth 5s 4d per gallon ?
48 8 + 2 = 10
60 8 + 2 = 10 Here 64
66 16 + 4 = 20
-72 16 + 4 = 20 Then 10 : 10 :: 3 : 3
10 : 20 :: 3 : 6
10 : 20 :; 3 : 6
Ans. 3 gallons at 5s, 6 at 5s 6d, and 6 at 6sa 2. A grocer would mix teas at 125, 10s, and 6s per lb, with 20lb at 4s per lb. how much of each sort must he take to make the composition worth 8s per lb ?
Ans. 20lb at 4s, 10lb at 6s, 10īb at 10s, and 20lb at 12s. 3. How much gold of 15, of 17, and of 22 caracts fine, must be mixed with 5 oz of 18 caracts fine, so that the composition may be 20 caracts fine?
Ans. 5 oz. of 15 caracts fine, 5 oz of 17, and 25 of 22,
* In the very same manner questions may be wrought when several of the ingredients are limited to certain quantities, by finding first for one limit, and then for another. The two last Rules can need no demonstration, as they evidently result from the first, the reason of which has been already explained.
Position is a method of performing certain questions, which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of false numbers, to work with the same as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes also called Trial-and-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.---Position is either Single or Double.
SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their result proportional to their suppositions, belong to Single Position : such as those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. The rule is. as follows:
TAKE or assume any number for that which is required, and perform the same operations with it, as are described or performed in the question. Then say, As the result of the said operation, is to the position, or number assumed; so is the result in the question, to a fourth term, which will be the number sought*.
* The reason of this Rule is evident, because it is supposed that the results are proportional to the suppositions,
Thus, na : Q :: nz : 2,