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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 com. posed of those materials.
RULE. As the whole composition : is to the whole value : : 80 Is any part of the composition : to its mean price.
EYAMPLES. 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 the mixture is worth bu. ets. $cts.
bu. $ cls.
54)25,38(,17 Ans. 54 25,38 % If 20 bushels of wheat at I dol. 35 cts.
bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of the mixture be worth ?
Ans. $1.20 cts. 3. A Tobacconist mixed 36lb. of Tobacco, at Is. 6d. per Ib. 12 lb. at 25. a pound, with 12 lb. at 1s, 10d. per Ib. ; what is the price of a pound of this mixture ?
Ans. 13. 8d. 4. A Grocer mixed 2 C. of
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. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons, at 6s. 3d. ; what is a gallon of this composition worth
Ans., 5s. 10d. 23qrs.
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 together; I deniand the price oi 3} cwt. of this mixture ?
Ans. $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. 13pwt. 8gr. 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 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 Me: dial, 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 mean price in the like name, at the left hand.
2. Connect, or link, the price of each simple or ingredient, which is less than that of tậe 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.
3. Place the difference, between the mean price (or mixture rate) and that of each of the simples, opposite to the rạtos 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.
1. A merchant has spices, some at 9d. per lb. some at Is. some at 2s. and some at 2s. 6d. per lb. how much of each sort must he mix, that he may sell the mixture at 1s. 8d. per pound? d. d. 16,
d. 16. 9 10 at 9
4 12 | Gives the d. 12 10 2024)
8 2. A grocer would mix the following quantities of gugar; viz. at 10 cents, 13 cents, and 16 cents per lb. ; what quantity of each sort must be taken to make a mixture worth 12 cents per pound ? Ans. 5lb. at 10cts. 216. at 13cts. and 21b. at 16cts. pcr
16. 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 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 quantitity of each must he take,
Ans. 2 of cach 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 O 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 rúm at 58. 1 at 78. 6 do. at 9s. and 3
gals, woter. Or, 3 gals. rum at 5s, 6 do. at 7s. 1
do. al 9s, and i gal. water. 6. A grocer hath several sorts of sugar, viz. 1 sort at 12cts. per lb. another at 11cts. a third at 9cts. and a fivurth at 8cts. per lb. : I demand how much of each sort must he mix together, that the whole quantity may be afforded at 10 cts. per pound ?
cts. 2 at 12
1 at 12 1 at 11
2 at 11 1st Ans. 2d Ans.
3d Ans. 1 at 9
2 at 9
(1 at 8
16. cis. 3 at 12 2 at 11 2 at 9 3 at 8
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 ine mean rate, and place them alternately as in Case 1. 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.
1. A farmer would mix 10 bushels of wheat, at 70 cents 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 38 cents; what quantity of each must be taken.
70-8 stands against the given 48 2
[quantity. Mean rate, 38
2 : 24 bushels of rye. As 8 : 10 :: 10 : 121 bushels of corn.
32 : 40 bushels of barley.
These four answers arise from as many various ways of linking the ratco 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 composc this darwer, will likewise satisfy ui conditions of the question.
2. How much water must be mixed with 100 gallons of pam, worth 7s. 6d. per gallon, no reduce it to 6s. 3d. per gallon ?
Ans. 20 gallons. 3. A farmer would mix 20 bushels of rye, at 65 cents per bushel, with barley at 51 cts. and oats at 30 cts. per bushel ; how much barley and oats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cts. per bushel ?
Ans. 20 bushels of barley, and 61 ji bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other rum at 68. 8d. per gallon, and some water ; then I found it stood me in 6s. 4d. per gallon; I demand how much rum and how much water I took ?
Ans. 95 gals. rum at 6s. 8d. and 30 gals. water.
When the whole composition is limited to a given quantity.
Place the difference between the mean rate, and the seGeral prices alternately, as in Case I.; then, As the sum of the quantities, or difference thus determined, is to the given quantity, or whole composition : so is the difference of éach rate, to the required quantity of each rate.
1. A grocer had 4 sorts of tea, at 1s. 35. 6s. and 10s. per lb. the worst would not sell, and the best were too dear; he therefore mixed 120 lb. and so much of each sort, as to sell it at 4s. per lb. how much of each sort did he take? 26.
6 : 60 at 1 :)
Ib. b. 2 : 20 3 1 A, 12: 120 : : 1: 10 6