Sect. 2. Of Alligation Alternate. Alligation Alternate, is that by which the particular Quantities of every Ingredient concerned in any Mixture are found; when the particular Rates of every one of thofe Ingredients, and the mean Rate are given; being (as it were) the Converse to Alligation Medial; as will appear by the following Operations, which admit of three Cafes. Cafe I. The Particular Rates of any Ingredients proposed to be mixed, and the Mean Rate of the whole Mixture being given. To find how much of each Ingredient is requifite to compofe the Mixture; when the whole Quantity, or any Part thereof, is not limited. Queft. 1. How much Wheat at ss. the Bufhel, and Rye at 3 s. 6 d. the Bufhel, will compofe a Mixture that may be fold for 4 s. 4 d. the Bufhel? Note, In all Queftions of this Nature, it will be convenient to place the Mean Rate fo, as that it may be eafily compared with the particular Rates, in order to find every one of their Differences from the Mean Rate, by Inspection only. Thus, the Mean Rate=52 d. {} S Wheat 60 d. Then take the feveral Differences between the Mean Rate, and the Particular Rates; fetting down thofe Differences Alternately, and they will be the Quantities required. Thus 52 {60}{ That is 52 60 10=52 4210 for the Quantity of Wheat. 52 8 for the Quantity of Rye, that will com pofe the Mixture required. Add { The Proof by Alligation Medial. 10 Bufhels of Wheat at 60 d. per Bufhel = 600 d 336 d =936 di Then 18936: 1:52 d. 4 s. 4 d. the Mean Rate. Note, Altho' 10 and 8 do answer the Question, as plainly appears by the Proof; yet they are not the only two Numbers; for this Queftion, and all others of this Kind, will admit of various Answers, and all whole Numbers; for any two Numbers that are in the fame Proportion to one another, as 10 is to 8, will as truly answer the Question. Queft. 2. A Grocer would mix three Sorts of Tobacco together, viz. One Sort of 18 d. per lb, another Sort of 22 d. per lb, and a third Sort of 2 s. the lb. How much of each Sort, must he take, that the whole Mixture may be fold for 20 d. the Pound? Having fet down the given Rates, as bebre, the find each of their Differences from the propofed Mean Rate, and place those Differences alternately. Thus, Mean Rate 20 18 2 5 4 + 2 = 24 {:} 24 2=2018 20 and 2220 Thefe Differences, 6. 2. 2 are the Quantities required. Their Value = 200 d. 10= the Number of Pounds. Then 10) 200 (20 the Mean Rate. Or indeed any three Numbers that have the fame Ratio to one another as 6 and 2 have, will answer the Queftion. 1 But if only one of the three given Rates had been greater than the Mean Rate; as fuppofe 14 d. per Pound, 18 d. per Pound, and 24 d. per Pound, and the Mean Rate 20 d. as before. Then their Differences must have been placed, Queft. 3. A Vintner would make a Mixture of Malaga, worth 78. 6 d. per Gallon, with Canary at 6 s. 9 d. per Gallon, Sherry at 5 s. per Gallon, and White Wine at 4 s. 3 d. per Gallon; What Quantity of each Sort muft he take, that the Mixture may be fold for 6 s. per Gallon? In all Queftions of this Kind, wherein it is required to mix four Things together, two of them having their Prices greater, and two leffer than the mean Kate: you must always alligate or compare compare a greater and leffer Price with the mean Price, fetting down their Differences alternately, as in the first Example of this Section. of White will compof the Mixture required. Malaga 90 d. 12 Malaga Sherry 60 d. d.18 18 Sherry Or thus, 72 will, &c. Either of these Mixtures equally anfwer the Question, which may be easily tried as before in the last, &c. Cafe II. The particular Rates of all the Ingredients proposed to be mixed, the Mean Rate of the whole Mixture, and any one of the Quantities to be mixed being given: Thence to find how much of every one of the other Ingredients is requifite to compofe the Mixture. Note, This is ufually called Alligation Partial. Queft. 4. How much Wheat at 5 s. the Bufhel, must be mixed with 12 Bufhels of Rye at 3 s. 6 d. a Bufhel; that the whole Mixture may be fold for 4 s. 4 d. the Bufhel? In this Cafe you must fet down all the particular Rates, with the Mean Rate, and find their Differences juft as before; without any regard had to the Quantity given. Thus, Mean Rate 52 d. {ye Then SWheat 60 d. As the Quantity found by the Differences of the fame Name with the Quantity given: Is to the Quantity given :: So is any of the other Quantities found by the Differences : To the Quantity of it's Name. Thus 8: 82: 10: 15, the Quantity or Number of Bushels of Wheat required. Queft. 5. How much Malaga at 7 s. 6 d. the Gallon, Sherry at 5 s. the Gallon, and White Wine at 4 s. 3 d. the Gallon, must be mixed with 18 Gallons of Canary at 6 s. 9 d. the Gallon; that the whole Mixture may be fold for 6 s, the Gallon? The The Terms being fet down, &c. as before, will ftand }{ That is, 31 Gallons of Malaga, 27 of White Wine, and 13 of Sherry, being mixed with 18 Gallons of Canary, will make the Mixture required. Malaga 901 2 Or thus, 72 Canary 8121 White 519 Then 514) 37021 (72 d. — 6 s. Therefore the Quantities are as truly affigned here, as in the laft Work. Cafe III. The particular Rates of all the Ingredients proposed to be mixed; and the Sum of all their Quantities with the Mean Rate of that Sum being given; to find the particular Quantities of the Mixture. This is called Alligation Total, and is thus performed. Set down all the particular Rates, with the Mean Rate, and find their Differences, as before: add together all the Differences into one Sum; Then As the Sum of all the Differences: Is to the Sum of all the Queft. 6. Let it be required to mix Wheat at 5 s. the Bufhel, with Rye at 3 s. 6 d. the Bufhel; fo that the whole Quantity may be 27 Bufhels, to be fold for 4s. 4 d. a Bufhel; what Quantity of cach must be taken to make up the Mixture? Q 2 Mean Question 7. Suppose it were required to mix Malaga at 7 s. 6 d. the Gallon, with Canary at 6 s. 9 d. the Gallon; Sherry at 5 s. the Gallon, and White Wine at 45. 3 d. the Gallon; fo that the whole Mixture may be 90 Gallons; to be fold for 6s, the Gallon: How much of each fort will compose that Mixture ? Mean Rate 72 d. Malaga 9021 9 : 13 /1/20 White Wine. Either of thefe Ways do equally answer the Question, as may be eafily tried by Alligation Medial. As before, &c. Note, The Work of thefe Proportions may be much shortened (efpecially when there are many Ingredients to be mixed) if you obferve the fame Method as was propofed in the Rule of Fellowship, page 99, &c. I have made Ufe of the very fame Examples both in Alligation Medial, and Alternate, throughout the three Cafes; being, as I prefume, much better than if they had been different ones; becaufe the Learner may (if he confider them a little) eafily perceive, not only the Difference between the two Rules, but also wherein the |