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5. Three merchants A. B. and C. in partnership together for a year, put into one common stock as follows, viz. A. put in 4007. and at the end of 6 months withdraws 2001. B. puts in 3601. and at the end of 7 months 100/. more, but at the end of 9 months he takes out 1207. C. puts in 1907. and at the end of 8 months 1107. more, but at the end of 10 months he takes out 1007. they gain 4607. what is each man's share?

1. 8. d. qrs.
156 10 5 920 A's

1058

Answer

193 18 3

0,552 B's

1058

share.

109 11 3

2.644 C's

1058

ALLIGATION.

Alligation serves to solve questions that relate to the mixing of simples, and is either medial or alternate.

ALLIGATION MEDIAL.

Alligation Medial, from the rates and quantities of the simples given, discovers the rate of the mixture.

so is

RULE-Multiply each quantity by its rate: then say as the sum of the quantities: is to the sum of the products: any part of the mixture to the rate or value of that part.

:

EXAMPLES.

:

1. A grocer mingleth several sorts of sugar as follows, viz: 75 lb. at 10 cts. per lb.; 80lb. at 11 cts.; 90 lb. at 13 cts. and 55 lb. at 14 cts. What is a pound of this mixture worth?

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As 300

lb.

: 3570 : : 1 : 11 cts. Answer.

2. A vintner mingled 15 gallons of Canary, at 8s. per gallon, with 20 gallons of Malaga, at 7s. 6d. per gallon, with 10 gallons of Lisbon, at 6s. 4d. and 24 gallons of French white wine at 4s. I demand the price of this mixture per gallon? Ans. 6s. 2d. 248 grs.

3. A refiner having 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 oz. of 6 oz. fine, which he melts together, required the fineness of 1 lb. of this silver?

Ans. 6 oz. 13 dwts. 8 gr. Medicines are sometimes mixed or compounded by this rule; for understanding the manner whereofit will be neces sary to observe-That medicines, drugs, or simples, with

respect, to their qualities, are divided into five sorts, viz. hot, cold, dry, moist, temperate; and all, except the last, admit. of four degrees, represented by indices, as in the following table:

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Indices.
Degrees 4 3

I cold & moisture. Item. heat & dryness. The index 3 denotes cold or most in the 2d. degree, 9 denotes hot or dry in the 4th degree, and 5 denotes temperate.

4. Admit four simples, A. B. C. D. to be mixed as follows, viz: 4oz. of A. hot in the 4th degree; 1 oz. of B. hot in the 2d.; 1 oz. of C. temperate, and 3 oz. of D. cold in the 3d. degree. What is the quality of the mixture or compound?

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As 9

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Answer.

16 hot in the 1st degree. 5. Suppose four simples, E. F. G. H. to be mixed as follows, viz: 3 oz. of E. dry in the 4th degree; 2 oz. F. dry in the 1st. 4oz. of G. moist in the 3d, and 1 oz. of H. moist in the 2d degree. Quere the quality of the mixture ? Answer, temperate.

ALLIGATION ALTERNATE.

Alligation alternate, being the converse of alligation medial from the rates of the simples, and rate of the mixture given, finds the quantities of the simples.

RULE 1.-Place the rate of the mixture on the left side of a brace, as the root; and on the right of the brace set the rates of the several simples under one another, as the branches.

2. Link or alligate together the branches, so that every one less than the root, be linked with some one greater, or with all that are greater, and every one greater than the root, be linked with one less, or with all that are less.

3. Set the difference betwixt the root and the several branches, right against their respective yoke-fellows.-These alternate differences are the quantities required, which quantities are the answer for that rate, against which they stand.

EXAMPLES.

6. To mix gold of 18 carats fine, with that of 23 carats

fine, of 19, and of 16 carets fine, so that the composition may be 20 carats fine. What quantity of each must be taken? Ans. 3 each of 16, 18, 19, and 7 of 23.

7. A merchant would mix wines at 17s. 18s. and 22s. per. gallon, so that the mixture may be worth 20s. per gallon What quantity of each must be taken?

Ans. 2 at 17s. 2 at 18s. and 5 at 22s. 8. I desire to know how much gun-powder at 36 cts. 27 cts. and 24 cts. per lb. will compose a mixture worth 30 cents per lb. Ans. 9 lb. at 36 c. and 6 lb. each at 27 c. and 24 cts.

9. It is required to mix tea at 7s. per lb. with some at 8s. 6d. and 5s. 10d. per lb. so that the mixture may be at the rate of 6s. 104. per lb?

Ans. 12 lb. each at 7s. and 8s. 6d. and 21 lb. at 5s. 10d. 10. How much brandy at 14s. per gallon, and water must be mixed together, so that it may be worth 10s. per gallon? Ans. 10 gallons brandy, and 4 of water.

11. An apothecary would mix four simples, viz. A. hot in the 4th degree; B. hot in the 2d; C. temperate, and D. cold in the 3d, so as the mixture may be hot in the first degree. What quantity of each simple must he take?

9

1 of A| 1x9= 9 The branches may be alligated 4X7 28 other 6 ways, which would give 15 the like number of other answers.

4 of B

3 of C
1 of D

3x5

1x2= 2

6x9 54 proof. 54

12. Admit an apothecary to mix four simples, viz: E. dry in the 4th degree; F. dry in the 1st; G. moist in the 3d, and H. moist in the 2d. Required the quantity of each simple so as to make the mixture temperate?

Ans. 3 oz of E. dry in the 4th. 2 oz. of F. dry in the 1st. 4 oz. of G. moist in 3d. and 1oz. of H. moist in the 2d. degree.

ALLIGATION PARTIAL.

Alligation partial is when the price of each simple is given, also the quantity of one or more of them, and the mean rate, to find the several quantities of the rest.

RULE-Link or alligate the branches, and take the difference between each of them and the root, as in the last rule: then say, As the difference standing against the price of which the quantity is given, is to the said given quantity, so is any other difference, to the quantity required.

EXAMPLES.

13. How much tea at 6s. 6d. 7s. 6d. and 9s. per lb. must be

taken to be mixed with 36 lb. at 12s. per lb. that the mixture may be worth 8s. per lb ?

Ans. 108 lb. at 9s. 288 lb. at 7s. 6d. and 72 lb. at 6s. 6d. 14. How much alloy, and how much gold of 21 and 23 carats fine, must be melted with 30 oz. of 20 carats fine?

Ans. 163 oz. alloy, 30 oz. of 21, and 30 oz. of 23 carats fine. 15. How much brass at 28 cts. and pewter at 21 cts. per lb. must I melt with 50lb. of copper worth 32 cts. per lb. so that the whole mixture may stand in 24 cents per lb.

Ans. 50 lb. at 28 cts. and 200 lb. at 21 cts. 16. Suppose it required to reduce 63 gallons of brandy, at $1 30 cts. per gallon, to the price of $1 8 cts. per gallon? What quantity of water would it take?

90 Ans. 12 gallons.

17. A merchant would mix 3 sorts of wine of different prices, viz: one at 6s. per gallon, another at 8s. a third at 11s. with 49 gallons at 15s. per gallon, that the mixture may stand in 13s. per gallon: How much wine at 6s. 8s. and 11s. per gallon will it take? Ans. 7 gallons of each sort.

18. An innkeeper would, with 72 bushels of his best grain, at 80 cts. per bushel, mix other grain at 64 cts. at 48 cts. and 32 cts. per bushel: What quantity of the last three qualities must he take, that the mixture may be worth 56 cents per bushel ?

Answers.

(24 bushels each at 64 and 48 cts.72 bush. at 32 cts.
216 bushels each at 64 and 48 cts. 72 bush. at 32 cts.
18 bush. at 64, 72 bush. at 48, and 54 bush. at 32 cts.
96 at 64, 24. at 48, and 96 bush. at 32 cts.
54 bushels each at 64 and 48 cts. 72 bush. at 32 cts..
288
at 64 and 48 cts. 72 bush. at 32 cts.

72 bushels of each sort.

19. How much gold of 15 and 17 carats fine, must be mixed with 5 oz. of 18, and 13 oz. of 22 carats fine, that the composition may be 20 carets fine? Ans. 2 oz. of each sort.

N. B. Previous to the operation, the two given quantities must be supposed to be mixt by themselves; and the rate found by alligation medial.

20. A wine merchant would, with 10 gallons of wine, at 12s. per gallon, and 20 gallons at 15s. per gallon, mix other wines, at 188. and at 20s. per gallon: What quantity of the last two must he take that the mixture may be worth 16s. per gallon? Ans. 10 gallons of each sort.

ALLIGATION TOTAL..

Alligation Total, is when the price of each simple is given, as also the mean rate, and the proposed quantity of the compound, to find how much of each sort will make that quantity

:

RULE-After linking the branches, and taking the differences as before, say, as the sum of these differences is to the proposed quantity of the compound : : so is each particular difference: to its particular quantity.

EXAMPLES.

21. A vintner would mix wines at 8s. and 7s. per gallon, with cider at 1s. per gallon, so that the mixture may contain 25 gallons, and be worth 5s. per gallon: What quantity of each must he take?

Ans. 8 gal. at 8s. 8 at 7s. and 10 at 1s. per gallon. 22. A retailer has by him 4 sorts of tea, viz: of 5s. 6s. 8s. and 9s. per lb. out of these, he is ordered to make up a chest containing neat a hundred and a half, so as to be worth 7s. per lb. In what proportion must those teas be taken?

Ans. 28 lb. at 58. 56 lb. at 6s. 56 lb. at 8s. 28 lb. at 9s. per lb. 23. A goldsmith has several sorts of gold, viz. No. 1 of 24 carats fine, No. 2 of 22 carats fine, No, 3 of 18 carats fine, and No. 4 of 16 carats fine; How much of each sort must he take, to make a mass of 60 ounces, of 21 carats fine. Ans. 25 oz. of 24 carats, 15 oz. of 22 carats, 5 oz. of 18 carats,

15 oz. of 16 carats fine.

24. A druggist had three sorts of indigo, one sort was worth 48 cents per lb. another 60 cts. and another 96 cts. and out of these he made up a parcel of 21 lb. at 72 cents per lb. and an-. other parcel of 35 lb. at 84 cts. per lb: how much of every sort did he put in each parcel?

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96S

5
25 S

$487

60 per lb. Ans. · 5 at 60 per lb.

(96

25.Brandy at 7s. and 11s. 6d. per gallon, is so to be mixed that a hhd. of 63 gallons may be sold for 251. 4s. Quere the quantity of each sort?

Ans. 49 gallons at 7s. 14 gallons at 11s. 6d. per gallon. 26. A silversmith has silver of 113 oz. fine, and of 7 oz. fine and has orders to make up a piece of work, requiring 35 lb. of 9 oz. fine; how much must he take of each ? Ans. 21 lb. 4 oz. 13 dwt. 8 grs. of 111⁄2 oz. 13 lb. 7oz. 6dwt. 16 gr. of 7 oz. fine.

27. A grocer has three sorts of sugar, viz. of 12d. 11d. and 9d. perib. now he wants to fill a tierce of 600 lb. so that he may sell it at 10d. per lb: what quantity of each must he ~take?

Ars. 120lb. at 12d.-120lb. at 11 d.-360lb. at 9d. per lb. 28. Suppose 9ib. of pure gold immersed in a vessel full of water to expel 3 lb. of water, 9 lb. of pure silver to expel 6lb. of water, and 9 lb. of a mass made up of gold and silver to

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