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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 4001. and at the end of 6 months withdraws 2001. B. puts in 3601. and at the end of 7 months 1001. more, but at the end of 9 months he takes out 1201. C. puts in 1901. and at the end of 8 months 1101. more, but at the end of 10 months he takes out 1001. they gain 4601. what is each man's share?

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8. d. qr8.

156 10. 5 0,920 A's
Answer 193 18 3 0 5.52 B's share.

109 11 3 2-644 C's

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ALLIGATION. Alligation serves to solve questions that relate to the mixa ing 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.

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. ; 80 lb. at 11 cts. ; 90 lb. at 13 cts. and 55 lb. at 14 cts. What is a pound of this mixture worth ?

1b. cts. 75x10= 750 80 X11= 880 90x13=1170 55 X14 770

16. As 300 3570 : : 1 113 cts. Answer.

2. A vintner mingled 15 gallons of Canary, at 8s. per gallon, with 20 gallons of Malaga, at 78. 6d. per gallon, with 10 gallons of Lisbon, at 68. 4d. and 24 gallons of French white wine at 45. I demand the price of this mixture per gallon ?

Ans. 68. 2d. 249 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. 602. 13 dwts. 8 gr. Medicines are sometimes mixed or compounded by this rule ; for understanding the manner whereof it will be neces. sary to observe-That medicines, drugs, or simples, with

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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 :

Indices. 2 3 4 5 6 7 8
Degrees 3

0

3 4 I coid & 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 : 40z. 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?

A. 4x9=36
B. 1x7=7
C. 1x5= 5
D. 3X2= 6

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As 9 54 :

:1:6 hot in the 1st degree. Answer. 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. : 402. 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 aligate 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 betwist the root and the several bran. ches, right against their respective yoke-fellows.- These alternate differences are the quantities required, which quan. tities are the answer for that rate, against which they stand.

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

per lb.

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 17 s. 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

Ans. 9 lb. at 36 c. and 6 ib. 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. 10.11. per

lb? Ans. 12} lb. each at 78. and 8s. 6d. and 21 lb. at 5s. 10d.

10. How much brandy at 14s. per gallon, and water must be mised 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 dea free. What quantity of each simple must he take ?

1 of A 1x9= 9 The branches may be alligated
4 of B 4X7=28 other 6 ways, which would give
3 of c
1 of D 1X2= 2

|ix5=13 the like number of other answers.

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 ? Liis. 3 oz of E. dry on the 4th. 20%. of F. dry in the 1st. 4 02. of G. moist in 3d. and 1 02. of H. moist in the 2d. degree.

ALLIGATION PARTIAL. Alligation partial is when the price of each simple is givell, also the quantity of one or more of them, and the mean: Tate, 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 wiiich the quantity is given, is to the said given quantity, so is any other difference, to the quantity required.

EXAMPLES.
How much tea at 6s. 6d. 7s.6d. and 9s. per Ib. must be

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

Ans. 108 lb, at 9s. 288 lb. at 78. 6d. and 72 lb. at 6s. 6d.

14. How much alloy, and how much gold of 21 and 23 ca.. rats fine, must be melted with 30 oz. of 20 carats fine ?

Ans. 16 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 50 lb.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 gailon, to the price of $1 8 cts. per gallon? What quantity of water would it take ?

Ans. 1210. gallons. 17. A merchant would mix 3 sorts of wine of different prices, viz : one at 68. per gallon, another at 8s. a third at 11s. with 49 gallons at 158. per gallon, that the mixture may stand in 138. 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 otlier grain at 64 cts. at 48 cts.. and 32 cts. per bushel : · What quantity of the last three qualities must be take, that the mixture may be worth 56 cents per bushel ?

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. Answers 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 02.

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 şort.

ALLIGATION TOTAL, Alligation Total, is when the price of each simple is given, as also

mean rate, and the proposed quantity of the compound, to find how much of each sort will make that quantity

per lb.

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 78. per gallon, with cider at 1s. per gallon, so that the mixture may contain 25 gallons, and be worth 58. 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 58. 63. 8s. and 9s. per Ib. out of these, he is ordered to make up a chest containing neat a hundred and a half, so as to be worth 7s.

In what proportion must those teas be taken ? Ans. 28 lb. at 58. 56 lb. at ós. 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 43 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 ? ib.

lb. Ans.

> 60 per lb. Ans. at

() 96 S 25. Brandy at 7s. and 11s. 6d. per gallen, 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 78. 14 gallons at 11s. 6d. per gallon. 26. A silversnith has silver of 11.) 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 119 oz. 13 16.702. Odwt. 16 gr. of 7 oz.

3. fine. 27. A giocer has three sorts of sugar, viz. of 12d. 11d. and 9d. prib. 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 121.-12016. 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 61b. of water, and 9 lb. of a mass made up of gold and silver to

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