135796744(51,4 the root. 75)107 dividend. 132651=2d subtrahend. 7803) 31457=2d dividend. · 1357967443d subtrahend. 5x5x3=75 first divisor. 51×5TX51=152651 second subtranend. 51x51x3=7803 second divisor. 514X514×514-135796744 third subtrahend. 3. Required the sursolid, or fifth root of 6436343. 6436343)23-root. S2 2×2×2×2×580)323 dividend. NOTE.--The roots of most powers may be found by the square and cube roots only; therefore, when any even power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which reduces it to half the given power, then the square root of that power reduces 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 root of that reduces it to a sixth power: and the square root of a sixth power to a cube, EXAMPLES. S. What is the biquadratę, or 4th root of 19987175376? Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 Ans. 48. 464. 5. Extract the square, biquadrate, or 8th root of 72138 95789333336. 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 bushel, 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? 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, 20cts. 3. A Tobacconist mixed $6 lb. of Tobacco, at is. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 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 inixture? Ans. 7 13s. 5. A Wine merchant mixes 15 gallons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons, at 6s. Sd.; what is a gallon of this composition worth? Ans. 5s. 10d. 248grs. 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 sort at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. and he would mix an equal quantity of each togeth er; Idemand the price of 54 cwt. of this mixture? Ars. $34 12cts. 5m. 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. 6oz. 1Spwt. 8gr. fine. & Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what is 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 rates of all the ingredients are given without any limited quantity. RULE. 1. Place the several rates, or prices of the simples, being reduced to one denomination, in a column under each other, and the meas price in the like name, at the left kand. 2. Connect, or link, the price of each simple or ingredient, which is less than that of the mean rate, with one or any number of those, which are greater than the mean rate, and each greater rate, or price with one, or any number of the less. 5. Place the difference, between the mean price (or misture 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 Ib. some at ́s. some at 2s. and some at 2s. 6d. per ib. how much of ech sort must he mix, that he may sell the mixture at 1s Sd. per pand? 2. A grocar would mix the following quantities of su-gar; viz. at 10 cents, 13 cents, and 16 cts. per Ib.; what quantity of each sort must be taken to make a mixture worth 12 cents per pound? Ans. 5lb. at 10cts. 2lb. at 13cts, and 2lb. at 16 cts, për lb. 5. 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 composition 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 some of 19, 21, and 24 carats fine, so that the compound may 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. 7s. and 9s. per gallon, with water at 0 per gallon together, so that the mixture may be worth 6s. 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 at 9s. and 3 gals. water. Or, 3 gals. rum at 5s. 6 do. at 7s. 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 b.; I demand how much of each sort must he 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, 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 given, is to that quantity: so is each of the other differences, severally, to the several quantities required. EXAMPLES. 1. A farmer would mix 10 bushels of wheat, at 70 cts. per bushel, with rye at 48 cts. corn at 36 cts. and barley at 30 cts. per bushel, so that a bushel of the composition may be sold for $8 cents; what quantity of each must be taken. *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 |