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2. A merchant has Canary wine, at 3s. per gallon, Sherry, at 2s. Id. and Claret at Is. 5d. per gallon: How much of each sort must he take, to sell it at 2s. 4d. per gallon?

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Mean rate 28d. ? 25

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3. How much barley at 40c. rye at 60c. and wheat at 80c. per bushel, must be mixed together, that the compound may be worth 621c. per bushel?

Ans. 17 bushels of barley, 17 of rye, and 25 of wheat. 4. A goldsmith would mix gold of 19 carats fine, with some of 16, 13, 23 and 24 carats fine, so that the compound may be 21 carats fine: What quantity of each must he take?

Ans. 5oz. of 16 carats fine, 5oz. of 18, 5oz. of 19, 10oz. of 23, and 10oz. of 24 carats fine.

5. It is required to mix several sorts of wine, at 60c. 90c. and D.1 15c. per gallon, with water, that the mixture may be worth 75c. per gallon Of how much of each sort must the composition consist? Ans. 40 galls. of water, 15 galls. of wine, at 60c. 15 galls. do. at 90c. and 75 galls. do. at D.1 15c.

CASE II.

When the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture are given, to find the several quantities of the rest, in proportion to the quantity given.

RULE.

Take the differences between each price, and the mean rate, and place them alternately, as in Case 1. Then, as the difference standing against that simple, whose quantity is given, is to that quantity, so is each of the other differences, severally, to the several quantities required.

EXAMPLES.

1. A merchant has 40lb. of tea, at 6s. per lb. which he would mix with some at 5s. 8d. some at 5s. 2d. and some at 4s. 6d. : How much of each sort must he take, to mix with the 40lb. that he may sell the mixture at 5s. 5d. per lb. ?

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*Note, the 2d. and 3d. questions admit but of one way of linking, and fo but of one answer; yet all numbers in the same proportion between themselves, as the numbers, which compofe the answer, will likewife fatisfy the condition of thequestion.

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2. A farmer being determined to mix 20 bushels of oats, at 60c. per bushel, with barley, at 75c. rye, at D.1, and wheat, at D.1.25c. per bushel; I demand the quantity of each, which must be mixed with the 20 bushels of oats, that the whole quantity may be worth 90c. per bushel ? Ans. 70 of barley, 60 of rye, and 30 of wheat, (or 20

of each.)

3. How much gold of 16, 20 and 24 carats fine, and how much alloy, must be mixed with 10 oz. of 18 carats fine, that the composi tion may be 22 carats fine.

Ans. 10oz. of 16 carats fine, 10 of 20, 170 of 21, and 10 of alloy.

ALTERNATION TOTAL.*

CASE III.

When the rates of the several ingredients, the quantity to be compounded, and the mean rate of the whole mixture are given, to find how much of each sort, will make up the quantity.

RULE.

Place the differences between the mean rate, and the several prices alternately, as in Case ; then, as the sum of the quantities, or differences thus determined, is to the given quantity, or whole composition; so is the difference of each rate, to the required quantity of each rate. EXAMPLES.

1. Suppose I have 4 sorts of currants, at 8d. 12d. 18d. and 22d. per lb.; the worst will not sell, and the best are too dear; I therefore conclude to mix 120lb. and so much of each sort as to sell them at 16d. per lb.; how much of each sort must I take?

16d.

• To this Cafe belongs that curious question concerning king Hiero's crown. Hiero, king of Syracufe, gave orders for a crown to be made, entirely of pure gold; but fufpecting the workmen had debased it, by mixing with it filver or copper, he recommended the discovery of the fraud to the famous Archimides, and defired to know the exact quantity of alloy in the crown.

Archimides, in order to detect the impofition, procured two other maffes, one of pure gold, and the other of filver, or copper, and each of the fame weight with the former; and by putting each feparately into a veffel full of water,the quantity of water expelled by them, determined their fpecifick bulks; from which, and their given weights, it is easier to determine the quantities of gold and alloy in the crown by this case of Alligation, than by an Algebraick process.

Suppose the weight of each mafs to have been 5lb. the weight of the water expelled by the alloy, 230z. by the gold, 130z. and by the crown 160z. that is, that their fpecifick bulks were as 23, 13, and 16; then, what were the quantities of gold and alloy respectively in the crown?

Here, the rates of the fimples are 23 and 15, and of the compound 16, whence, 137 7 of gold? And the fum of thefe is 7+3=10, which should have 223 3 of alloy been but 5, whence, by the rule,

16

10 : 5 ::

2

57: 3lb. of 2004 the Answer,

1lb. of alloy

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2. A goldsmith has several sorts of gold; viz. of 15, 17, 20 and 22 carats fine, and would melt together, of all these sorts, so much as may make a mass of 40oz. 18 carats fine; how much of each sort is required?

Ans. 16oz. 15 carats fine, 8oz. 17, 4oz. 20, and 12oz. of 22 carats fine.

3. A merchant would mix 4 sorts of wine, of several prices, viz. at 75c. ID. 25e. 1D. 50c. and 1D. 624c. per gallon; of these he would have a mixture of 60 gallons, worth 7s. per gallon; what quantity of each sort must he have?

Ans. 8 at 75c. 16at ID. 25c. 40 at 1D. 50c. and 8 at ID. 62 c. Or, 16 at 75c. 8 at 1D. 25c. 8 at 1D. 50c. and 40 at ID. 62c.

4. How many gallons of water, of no value, must be mixed with wine, at 4s. per gallon, so as to fill a vesssl of 80 gallons, that may be afforded at 2s. 9d. per gallon?

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Find, by Alligation Medial, what will be the rate of a mixture made of the given quantities of the limited simples only; then, consider this as the rate of a limited simple, whose quantity is the sum of the first given limited simples, from which, and the rates of the unlimited simples, by Case 2d. calculate the quantity.

EXAMPLES.

1. How much wine, at 80c. and at 874c. per gallon, must be mixed with 8 gallons at 75c. and 12 galls. at 90c. per gallon, that the mixture may be worth 824c. per gallon?

8 gallons, at 75c.

Limited simples {12 gallons, at 90

-

20

Gal. D. c. Gal. c.

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As 20: 16 80 :: 1: 84 per gallon.

* The three last Cases need no demonstration, as the 2d. and 3d. evidently refult from the first, and the last, from Alligation Medial, and the second Cafe in Alternate.

2.Q

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Now, having found the rate of the limited simples, the question may stand thus: How much wine, at 80c. and 874c. per gallon, must be mixed with 20 gallons at 84c. per gallon, that the mixture may be worth 824c. per gallon?

1+56 gallons, at 80c.

84

87

22 gallons, at Soc. per gallon.}

_871

Answer.

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2. How much gold, of 14 and 16 carats fine, must be mixed with 6 oz. of 19, and 12 of 22 carats fine, that the composition may be 20 carats fine? Ans. 1 oz. of each sort.

POSITION.

POSITION is a rule, which, by false or supposed numbers, taken at pleasure, discovers the true ones required. It is divided into two parts; single and double.

SINGLE POSITION.

Single Position teaches to resolve those questions, whose results are proportional to their suppositions: such are 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 a certain proposed number of times.

RULE.-1. Take any number, and perform the same operations with it as are described to be performed in the question.

2. Then say, as the sum of the errours is to the given sum, so is the supposed number, to the true one required.

Proof. Add the several parts of the sum together, and if it agrees with the sum, it is right.

EXAMPLES.

The reafon of this rule is obvious, it being evident that the results are propor tional to the fuppofitions,

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EXAMPLES.

1. A school-master, being asked how many scholars he had, said, If I had as many more as I now have, three quarters as many, half as many, one fourth and one eighth as many, I should then have 435: Of what number did his school consist?

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2. A person lent his friend a sum of money unknown, to receive interest for the same at 6 per cent. per annum, simple interest, and, at the end of 12 years, received for principal and interest 860D. : What was the sum lent? Ans. D.500.

3. A, B and C joined their stocks, and gained D.353 124c. of which A took up a certain sum, B took up four times so much as A, and C, five times so much as B: What share of the gain had each ?

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4. A, B, C and D spent 35s. at a reckoning, and, being a little dipped, they agreed that A should pay 3, B4, C, and D 4: What

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5. A certain sum of money is to be divided between 5 men, in such a manner as that A shall have 4, B, C, D, and E the remainder, which is £.40: What is the sum ?

Suppose 2001. then +++ = 120.

20

200-120 80. As 80: 40 :: 200: 100 Ans. 6. A person, after spending and of his money, had 2631. left : What had he at first? Ans..160.

7. A and B, talking of their ages, B said his age was once and an half the age of A; C said his was twice and one tenth the age of both, and that the sum of their ages was 93: What was the age of each? Ans. A's 12, B's 18, and C's 63 years. A, B and C ; A can fill it in an hour, of an hour: In what time will they all Ans. hour.

8. A vessel has 8 cocks, B in 4 of an hour, and C in fill it together?

9. A person having about him a certain number of dollars, said that,,, and of them would make 57: Pray, how many had he?

Ans. 60.

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