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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 62 c. 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, 18, 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 60ĉ. 90c. and $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. 40galls. of water, 15galls. of wine, at 60c. 15galls. do. at 90c. and 75galls. do. at $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 I. 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 40 of tea, at 63. per b, 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 4015, that he may sell the mixture at 53. 5d. per ?

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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 $1, and wheat, at $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 10oz. of 18 carats fine, that the composition may be 22 carats fine.

Ans. 10oz. of 16 carats fine, 10 of 20, 170 of 24, 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 1; 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 3d. 12d. 18d. and 228. per ib; the worst will not sell, and the best are too dear; I therefore conclude to mix 120 and so much of each sort as to sell them at 16d. per th; how much of each sort must I take?

d.

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Sum=20

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120

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, Soz. 17, 4oz. 20, and 12oz. of 22 carats fine.

To this Case belongs that curious question concerning king Hiero's crown. Hiero, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but suspecting the workmen had debased it, by mixing with it 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, one of pure gold, and 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 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 Algebraic process. Suppose the weight of each mass to have been 5lb. the weight of the water expelled by the alloy, 23oz. by the gold, 13oz. and by the crown 16oz. that is, that their specifick 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 simples are 23 and 13, and of the compound 16, whence, of gold And the sum of these is 7+3=10, which should have 23 3 of alloy S been but 5, whence, by the rule,

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10:5 :: (7:31lb. of gold the Answer.

of alloy S

3. A merchant would mix 4 sorts of wine, of several prices, viz. at 75c. $1 25c. $1 50c. and $1 62 c. per gallon; of these he would have a mixture of 72 gallons, worth $1 374c. per gallon; what quantity of each sort must he have?

Ans. 8 at 75c. 16 at $1 25c. 40 at $1 50c. and 8 at $1 621c. Or, 16 at 75c. 8 at $1 25c. 8 at $1 50c. and 40 at $1 62 c.

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

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When more than one of the simples are limited.

RULE.

Ans.

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 II. calculate the quantity.

EXAMPLES.

1. How much wine, at 80c. and at 87 c. per gallon, must be mixed with 8 gallons at 75c. and 12 gallons at 90c. per gallon, that the mixture may be worth 82 c. per gallon?

Limited simples {

8 gallons, at 75c.= 6
12 gallons, at 90 = 10 80c.

20

Gal. $ c. Gal. c.

16 80

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

Now, having found the rate of the limited simples, the question may stand thus: How much wine, at 80c. and 87c. per gallon, must be mixed with 20 gallons at 84c. per gallon, that the mixture may be worth. 821c. per gallon?

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

W W

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

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?

The operations contained in the question being performed upon the answer or number to be found, will give the result contained in the question. The same operations, performed on any other number, will give a certain result. When the results are proportional to their supposed numbers, it is manifest that one result must be to the result in the question, n the supposed number is to the true one or answer. In any cases, when the results are not proportional to their suppositions, the answer cannot be found by this rule.

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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 $860: What was the sum lent? Ans. $500.

3. A, B and C joined their stocks, and gained $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? $14 12 c. A's share.

Ans.

56 50 B's share. 282 50 C's share.

4. A, B, C and D spent 35s. at a reckoning, and, being a little dipped, they agreed that A should pay, B, C, and D : What did each pay in the above proportion?

s. d. (A, 13 4

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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, B, C, D, and E the remainder, which is £40: What is the sum ? Suppose £200, then +++120.

200-120=80. As 80 40 :: 200: 100 Ans. 6. A person, after spending and of his money, had £263 left : 183 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.

8. A vessel has 3 cocks, A, B and C ; A can fill it in an hour, Bin of an hour, and C in of an hour: In what time will they all fill it together? Ans. hour.

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.

10. A Gentleman bought a chaise, horse and harness, for $500, the horse cost more than the harness, and the chaise more than the horse What was the price of each?

Ans.

Harness $127 65c. 927m.
Horse 159 57 42
Chaise 212 76 514

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