135796744(51,4 the root. 5)107 dividend. 132651=2d subtrahend. 7803) 31457=2d dividend. 135796744-3d subtrahend. 5x5x3=75 first divisor. 51×51×51=132651 second subtrahend. 2. Required the sursolid or 5th root of 6436343. 6436343(23 root. 32 2×2×2×2×5=80)323 dividend. 23 × 23 × 23 × 23× 23=6436343 subtrahend. NOTE. The roots of most powers may be found by the square and cube roots only; therefore, when any ever power is given, the easiest method will be (especially in very high power) to extract the square root of it, which re duces it to half the given power, then the square root o that power reduces it to half the same power; and so ol till you come to a square or a cube. For example: suppose a 12th power be given; the square root of that reduces it to a 6th power: and the square root of a 6th power to a cube. EXAMPLES. 3. What is the biquadrate, or 4th root of 19987173376 ? Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464. Ans. 48. 5. Extract the square quadrate, or 8th root of 72138 95789338336. Ans 96. ALLIGATION, IS the method of mixing several simples of different qualities, so that the composition may be of a mean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials. RULE. As the whole composition is to the whole value:: so is any part of the composition: to its mean price. EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushe, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a bushel of this mixture is worth? bu. cts. $cts. bu. $cts. 15 at 64-9,60 As 54: 25,38 : bu. : 1 54)25,38(,47 Ans. 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth? Ans. $1,20 cts. 3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at Is. 10d. per lb.; what is the price of a pound of this mixture? Ans. 1s. 8d. 4. A grocer mixed 2 C. of sugar at 56s. per C. and 1 C. at 43s. per C. and 2 C. at 50s. per C. together; I demand the price of 3 cwt. of this mixture? Ans. £7 13s. 5. A wine merchant mixes 15 gallons of wine at 4s. ad. per gallon, with 24 gallons at 6s. 8d. and 20 gallons at Os. 3d; what is a gallon of this composition worth? Ans. 5s, 10d. 247 grs. 6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sou at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt and he would mix an equal quantity of each together; I demand the price of 3 cwt..of this mixture? Ans. $34 12 cts. 5 m. 7. A goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine; pray what is the quality or fineness of this composition? Ans. 6 oz. 13 pwt. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what i the quality or fineness of this mass? Ans. 19 carats fine. ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Medial, and may be proved by it. CASE I When the mean rate of the whole mixture, and the rate, of all the ingredients are given, without any limited quar tity. RULE. 1. Place the several rates, or prices of the simples, be ing reduced to one denomination, in a column under each other, and the mean price in the like name, at the left hand 2. Connect, or link the price of each simple or ingredi ent, which is less than that of the mean rate, with one of any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num ber of the less. 3. Place the difference, between the mean price (or mig, ture rate) and that of each of the simples, opposite to the rates with which they are connected. 4. Then, if only one difference stands against any rate, It will be the quantity belonging to that rate, but if there be several, their sum will be the quantity. EXAMPLES. 1. A merchant has spices, some at 9d. per lb. some at 1s. come at 2s. and some at 2s. 6d. per lb. how much of each Fort must he mix, that he may sell the mixture at 1s. 8d. per pound? d. lb. d. d. lb. 2. A grocer would mix the following qualities of sugar; riz. at 10 cents, 13 cents, and 16 cents per lb.; what quanity of each sort must be taken to make a mixture worth 42 cents per pound? Ans. 5 lb. at 10 cts. 2 lb. at 13 cts. and 2 lb. at 16 cts. per lb. 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. per lb. how must he mix them so as to afford the composiion for 12s. per lb. ? Ans. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with ome of 19, 21, and 24 carats fine, so that the compound nay be 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, viz. at 5s. Ys. and 9s. per gallon, with water at 0 per gallon, togeher, so that the mixture may be worth 6s. per gallon; how inuch of each sort must the mixture consist of? Ans. 1 gal. of rum at 5s., 1 do. at 7s., 6 do. at 9s. and 3 gals. water. Or, 3 gals. rum at 5s., 6 do. at 7s., 1 do. at 9s. and I gal. water. 6. A grocer hath several sorts of sugar, viz. one sort at 12 ts. per lb. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per lb. ; I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10 cents per pound? ALTERNATION PARTIAL, Or, when one of the ingredients is limited to a certain quantity, thence to find the several quantities of the rest, is proportion to the quantity given. RULE. Take the differences between each price, and the mear rate, and place them alternately as in CASE I. Then, as the difference standing against that simple whose quantity i given, is to that quantity: so is each of the other differ ences, severally, to the several quantities required. EXAMPLES. 1. A farmer would mix 10 bushels of wheat, at 70 centr per bushel, with rye at 48 cts. corn at 36 ets. and barley a 30 cts. per bushel, so that a bushel of the composition may be sold for 38 cts.; what quantity of cach must be taken? 8 stands against the given quan 70 48 Mean rate, 38 36 30 2 10 32 2: 2 bushels of rye. [tity As 8: 10 :: 10 12 bushels of corn. These four answers arise from as many various ways of linking the rates of the ingredients together. Questions in this rule admit of an infinite variety of answers: for after the quantities are found from different methods of linking; any other numbers in the same proportion between themselves, as the numbers which compose the answer, will likewise satisfy the conditions of the question. |