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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 fir.d the mean price of the mixture composed of those materials.
RULE. A.s the whole composition : is to the whole value : : 90 is any part of the tomposition : 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, ana 21 bushels of oats, at 28 cts. a bushel ; I demand what a bushel of this mixture is worth?
br. cts. Scts. bu. S cts. bu.
54)25,38(,47 Answer. 54 :
25,38 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 36 lb. of Tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per Ib.; what is the price of a pound of this mixture ?
to. Ans. Is. 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. 67 138.
5. A Wine merchant mixes 15 gallons of wine at 48. 2d. per gallon, with 24 gallons at 68. &. and 20 gallons, at os. 30.; what is a gallón of this composition worth?
Ans. 5s. 10d. 24 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 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 demand the price of 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. &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 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 simpies, 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 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.
3. Place the difference, between the mean price for mixture 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.
1. A merchant has spices, some at 9d. per
Ib. some at 1s. some at 2s. anıl some at 2s. 6d. per 1b. how much of each sort must he mix, that he may sell the mixture at 1s. 8d. per pound ? d. d. 1b
lb. 9m 10 at 9 d.
4 12 | Gives the d. 12 10 20 24 8 24 ? Answer. On 2024 11 30 11 30
302. A grocer would mix the following quantities of sugar; viz. at 10 cents, 13 cents, and 16 cts. per lb. ; 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 21b. at 16 cts. per lb.
3. A grocer has two sorts of tea, viz. at 9s. and at 15s. per Ib. 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. ? 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 O per gallon together, so that the mixture may be worth ős. 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 58. 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 cts. per lb. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per Ib. ; I demand how much of each sort must he mix together, that the whole quantity may be afforded at 10 cents per pound ?
Ib. , cts.
16. cts. 2 at 12
1 at 12
r3 at 12 1 at 11
2 at 11 1st. Ans.
2 at 11
2 at 9
3 at 8
CASE II. 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 inean 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 dif. ferences, severally, to the several quantities required.
70 8 stands against the given quan-
(tity. Mean rate, 385
2 : 24 bushels of rye. As 8: 10: 10 : 12 bushels of corn.
32 : 40 bushels of barley. * These four answers arise from as many
various ways of linking the rates of the ingredients together.
Questions in this rule admit of an infinitę variety of answers: for ufter the quantities are found from different methods of linkiny; any other numčers in the same propor. tion between theniselves, as the numbers, which compose the answer, will likewise satisfy the conditions of the question.
2. How much water must be mixed with 100 gallons of rum, worth 7s. 6d. per gallon, to reduce it to-68. 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 vats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cents per bushel?
Ans. 20 bushels of barley, and 614 bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other rum at 6s. 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. Sr. and so gals. water.
When the whole composition is limited to a given quantity.
Place the difference between the mean rate, and the several 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 diffepeace of each rate, to the required quantity of each rate.
1. A grocer had four sorts of tea, at 18. 39. 6s. and 10s. per
Ib. the worst would not sell, and the best were too dear; he therefore mixed 120 15. and so much of each sort, as to sell it at 4s. per lb. ; kow much of each sort did he take? S. 16.
76 : 60 at 3
2 26. lb. ย
2 : 20 3 6
lb. 1 As 12 : 120 : ; 1 : 10 6
13 : 30