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1. A merchant would mix wines at 168, at 18s, and at 228 per gallon, so as that the mixture may be worth 20s the gallon : what quantity of each musi be taken?
2 at 168 Here 20 18 2 at 188
4 + 2 = 6 at 228. Ans. 2 gallons at 168, 2 gallons at 188, and 6 at 228. 2. How much wine at 68 per gallon, and at 45 per gallon must be mixed together, that the composition may be worth 58 per gallon ?
Ans 1 qt or l gall, &c. 3. How much sugar at 4d, at 6d. and at ild per lb, must be mixed together, so that the composition formed by them may be worth 7d per lb ? Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity
of each sort. 4. How much corn at 28 6d, 38 8d, 4s, and 48 8d per bushel, must be mixed together, that the compound may be worth 38 100 per bushel ?
Ans. 2 at 28 6d, 2 at 3s 8d, 3 at 48, and 3 at 4s 8d. 5. A goldsmith has gold of 16, of 18, of 29, 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 128, wine at 108, cyder at ls, and water at O per gallon together, so that the mixture may be worth 88 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 ca. racts fine?
16 Then, as 16:40 :: 2 : 5
and 16 : 40 :: 10:25 Ans. 5 o z of 15, of 17, and of 18 caracts fine, and 25 oz of 22
Ex.2. A vintner has wine at 48, at 58, at 58 6d, and at 68 a gallon ; and he would make a mixture of 18 gallons, so that it might be afforded at 58 4d per gallon ; how much of cach sort must be take?
Ans. 3 gal. at 48, 3 at 58, 6 at 58 6d, and 6 at 68.
A great number of questions might be here given relating to the specific gravities of metais, &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 fall 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 crown may be determined.
Suppose the weight of each crown to be 10lb, and that the water expelled by the copper or silver was 921b, by the gold 521b, and by the compound crown 641b; 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 52
28 of gold And the sum of these is 12 + 28= 40, which should have been but 10 ; therefore by the Rule,
the answer. 40:10 :: 28 : 71b of gold
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 givelı, is to the rest of he differences be verally; so is the quantity given, to the several quantities required.
1. How much wine at 58, at 58 6d, and 68 the gallon, must be mixed with 3 gallons al 4.8 per gallon, so that the mixture may
be worth 58 4d per gallon ?
8 +2= 10
8 +2 = 10 Here 64
10 : 20 :: 3:46
Ans. 3 gallons at 5e, 6 at 58 6d, and 6 at 68. 2. A grocer would mix teas at 128, 108, and 68 per lb, with 30lb at 48 per Ib how much of each sort musi he take to make the composition worth 88 per lb?
Ans. 20lb at 48, 10lb at 68, 10lb at 108 and 20lb at 12s. 3. How much gold of 15, of 17, and of 22 caracts fine, must be mixed with 5 oz of 8 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 de monstration, as they evidently result from the first, the reason of whick has been already explained.
Position is a method of performing certain questions, which cannot be resolved by the common direct rules. soinetimes 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 num. bers, and hence 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 resoived 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 sught 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*.
is supposed that the
The reason of this Rule is evident, because results are proportional to the suppositions.
Thus, na: a :: ng : 2,
1. A person after spending and of his money, has yet remaining 601 ; what had he at Erst? Suppose he had at first 1201.
Proof. Now 1 of 120 is 40
fof 144 is 48 of it is 30
of 144 is 36
their sum is 70 which taken from 120
their sum taken from
leaves 60 as Then, 50 : 120 :: 60 : 144, the Answer. per question.
2. What number is that, which being multiplied by 7, and the product divided by 6, the quotient may be 21 ?
Ans. 18. 3. What number is that, which being increased by ), , and of itself, the sum shall be 75 ?
4. A general, after sending out a foraging } and of his men, had yet remaining 1000 ; what number had he in command ?
5. A gentleman distributed 52 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d, to each woman 4d, and to each child 2d : moreover there were twice as many women as men, and thrice as many children as women. How many were there of each ?
Ans. 2 men, 4 women, and 12 children?
have 6. One being asked his age, said, if it of the years lived, be multiplied by 7, and į of them be added to the product, the sum will be 219. What was his age ?
Ans. 45 years,