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135796744(51,4 the root.
125=ist subtraliene

75)107 dividend.

130651394 subtrahend. 7803) 31457=24 dividend..

135736744=sd subtrahend.

5 X5 X3=75 first divisor.
51 xölxöl=152651 second subtranend.
51X51X3=7803 second divisor.

5147514x514=135796744 third subtrahend. S. Required the sursolid, or fifth root of 6456343.

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2x2x2x2x5=80)393 dividend.

33x23x23x23x23=6436343 subtrahend. Note.--The roots of most powers may be found by the gequare and cube roots only; therefore, when any even power is given, the easiest method will be (especially in a very ligh power) to extract the square root of it, which rednies it tulall the given power, then the square root of that power reluces it to half the same power; and so on, till you come to a square or a cube.

For example: suppose a 12th power be given; the square foot of that reduces it to a sixth power: and the stjuare root of a sixth power to a cube,

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EXAMPLES.

S. Wirat is the biquadratę, or 4th root of 19987179376 ?

Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464.

Ans. 48. 5. Extract the square, biquadrate, or 8th root of 72138 95789388336.

Ans. 96

ALLIGATION, Is the method of mixing seyeral simples of different qualities, su that the composition niay be of ařinean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate.

ALL:GATION MEDIAL, Is wbeu the quantities and prices of scveral things are given, to find the mean price of the mixture composed vi those materials.

RULE As the whole composition : is to the whole value : : so is any part of the composition : tu its mean price.

EXAMPLES.

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1. A farmer mixed 15. bushels of rye, at 64 cents a bushel, 18 bushels of Inilian corn, at 55 cts. a busliel, and ci bushels of onts, it 28 cts. a busliel; I demand what a bushel of this mixture is worth?

cts. Sots. bu. S cts.
15 at 61=1,60 As 54 : 25,38 : : 1
18 *5 =9,00

1
2837,88

cis.

5+)25,58(,47 Answer. 54

25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel, be mixed with 10 busliels of rye at 90 cents per bushel, what wiil a bushel of this iliisture be worth?

Ans. $1, 20cts. 3. A Tobacconist mised 6 lb, of Tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at is. 10d. per Ib. ;. what is the price of a pound of this mixture?

Ans. 18. 82. 4. A Grocer mixed 2 C. of sugar, at 56s. per C. and 1 C. at 43s. per C. and 2 C. at 50s. por C. together; I demand the price of 3 cwt. of this inixture : Aris: 67 13s.

5. A Wine merchanť mixes 15 gallons of wine at 45. 2d. per gallon, with 24 gallons at 6s. Ed. and 20 gallons, at 69. sd.; what is a wallon of this composition worth?

Ans. 58. 30d. 2grs.

6. Agrocer hath several sorts of sugar, viz. onc sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort at 10 dols. per cwt. and a fourth sort at 1:2 dols. per cwt. and he would mix an equal quantity of each togellier; Idemand tire price of 35 cart of this mixture ?

ns. S34 12cts. 5m. 7. A Goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. tine, and 15 lb, of 6 oz. fine: pray what is the quality, or fineness of this coinposition :

Ans. 6oz. Ispuct. 8gr. fine. 8. Suppose 5 lb. of gold of 22 carats tine, 16. of 21 carats fine, and i lb. of alloy be melted together; what is the quality, or fineness of this massi

dus. 19 carats fine

ALLIGATION ALTERNATE, As the method of finding what quantity of each of the ingredients, whose rates are given, wili compose a misture of a given rate; so that it is the reverse of alligatio:: inedial, and may be proved by it.

CASE. 1. When the mean rate of the whole mixture, and the rates of all the ingredients are given without any limited quantity

RULE. 1. ??lace the several rates, or prices of the simples, being reduced to one denomination, in a column under each other, and thic mea price in the like name, at the left hand.

. Connect, or link, the price of each simple or ingreclient, which is less than that of the mean rate, with olie or any number of those, which are greater than the uncan rate, and each greater ratt, or price with one, or any mumber of the less.

3. Place the slitterence, between the mean price (or misture rate) and that of cach of the simples, opposite lu the rates with which they are connected.

4. Then, is only one difference stands against any rate, it will be the quantity belonging to that rate, but if there be several, their suni will be the quantity.

EXAMPLES

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1. À merchant has spices, some at 9d. per

Ib. s. Some at 2s. and some at 25. 6d. per ib. how much of e ch sort must he mix, that he may sell the mixture at 15 8d. per pand? kl. d. th.

d. 9- 10 at 97

Cา : 1. 4 12 ( Gires the 1.

10 20)

8 24 Pinszer, or 20 24 11
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30. Agrocar would mix the following quantities of sugar; viz. at 10 cents, 13 cents, and 16 cts. per

what quantity of each sort must be taken to make a mixture worth 12 cents per pound ? Ans. 516. at 10cts. 21b. et 13cts, and alh. at 16 cls. per lb.

3. grocer has two sorts of tca, viz. at 9s. and at 15s. per ib. how must lię mix them so as to afford the composition for - 128. per Ib.?

Ans. Ile must mix an equal quantity of each sort, 4. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound may bc 22 carats fine; what quantity of each must he take.

Ans. 2 of each of the first three sorts, and 9 of the last.

5. It is required to mix several sorts of rum, vize at 5s. Ts. and 9s. per gallon, with water at O per gallon together, so that the mixture may be worth os. per gallon ; how much of each sort must the mixture consist of: Ans. 1 gal. of Rum at 5s. 1 do. at 7s, 6 do nt Is. and 3

gals. Water. Or, S guls. rum at 5s. 6 do. at 75. 1

do, at 9s. and 1 gal. water. 6. A grocer hath several sorts of "sugar, viz.. one sort at 12 cts. per lb. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per H. ; I demand how much of each sort must he mix togetier, that the whole quantity may be altorded at 10 cents per pound ?

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lb. cts.

16.

cis, r 2 at 12

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3 at 12 1 at 11 2 at 11

2 at 11 Ist. Ang 2d Ans.

sd Ans.
1 at 9
2 at 9

2 at 9
2 at 8
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13 at 8 4th Ans. Slb. of each sort.*

CASE II. ALTERNATION PARTIAL. Or, when one of the ingredients is limited to a certain quantity, there to find the severalquantities of the rest, in proportion to the quantity given.

RULE. Take the difference 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 give!, is to that quantity : so is cach of the other dila ferences, severally, to the several quantities required.

TXAMPLES.

2

1. A farmer would mix 10 bushels of wheat, at To cts. per bushel, with rye at 48 cts. corn at 36 cts. and barley at so cts. per bushel, so that a bushel of the composition may be gold for 38 cents; what quantity of each inust be taken.

70- S stands against the given quan

487 Meall rate, 38

S6 10
30 S

2 : 25 busliels of ryc. As 8:10 :: 10 : 12) bushels of corn.

32 : 40 bushels of barley. * These four answers arise from as many kurious ways of linking the rates of the ingredients together.

Questions in this rule admit of an infinite variety of ansicers; for after the quantities are funnel from difförerit ariethods of linking ; any other ??umbers in the same proporrion betueen themselves, as the numbers which compose the unswer, will likewise satisfy the conditions of the questions,

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