Sect. 2. Of Alligation Alternate. ** Lligation Alternate, is that by which the particular Quantities 1 of every Ingredient concerned in any Mixture are found; when the particular Rates of every one of those 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 Cases. Case 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 requisite to compose the Mixture ; when the whole Quantity, or any Part thereof, is not limited. Queft. 1. How much Wheat at 5 s. the Bushel, and Rye at 3 s. 6 d. the Bushel, will compose a Mixture that may be sold for 4 s. 4 d. the Bushel ? Note, In all Questions of this Nature, it will be convenient to place the Mean Rate fo, as that it may be easily compared with the particular Rates, in order to find every one of their Differences from the Mean Rate, by Inspection only. S Wheat 60 d. 'Rye 42 d. Then take the several Differences between the Mean Rate, and the Particular Rates; setting down those Differences Alternately, and they will be the quantities required. 560 US10=52 — 42 52 242 8 = 60 --- 52 That is 52 — 42 = 10 for the Quantity of Wheat. And 6052 = 8 for the Quantity of Rye, that will compose the Mixture required. The Proof by Alligation Medial. 18the Number of Bushels. = 936 d. Then 18: 936 ::1:52 d. = 45. 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 Question, 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 ro is to 8, will as truly answer the Question. s 5: 4.2 15 I 12 Ces Viz. 10:8::3 20 : 167 &c. ad infinitum. 25 : 20 ) Quest. 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 tbird Sort of 2 s. the Ib. How much of each Sort, muft he take, that the whole Mixture may be fold for 20 d. the Pound? Having set down the given Rates, as be , the find each of their Differences from the proposed Mean Rate, and place those Differences alternately. Thus, (18764 +2=24 -- 20 and 22 - 20 Mean Rate 20 3 22.323 2018 (24) (2= 20-18 71082 Proof 2 lb at 22 d. the Pound come to 44 d. (2 lb at 24 d. (48) 10=the Number of Pounds. Their Value = 200 d. 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 Question. / s 9:32 That is, 6 : 2 :: 312 : 4€ (15:51 But if only one of the three given Rates had been greater than the Mean Rate; as suppose 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, (142 CA 124376 +25 Queft. 3. A Vintner would make a Mixture of Malaga, worth 75. 6 d. per Gallon, with Canary at 6 s. 9 d. per Gallon, Sherry at 55. 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 sold for 6 s. per Gallon? In all Questions 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 Rate: you must always alligate or compare compare a greater and lesser Price with the mean Price, setting down their Differences alternately, as in the first Example of this Section. r Malaga 90 d. 5 21=72-51 Thus, Mean Rate = 72 d. 3 Sherry bod, US 9=81 – 72 White 51 d. SU 18=90 LCanary 81 d. S. 112=72-60 Hence 21 Gallons of Malaga, 12 of Canary, 9 of Sherry, and 18 of White will compofc the Mixture required. r Malaga 90 d. I 5 12 Malaga Or thus. 22 ) Sherry 60 d. Sl18 Sherry Canary 80 d. l S 21 Canary (White 51 d. 1 9 White ) Either of these Mixtures equally answer the Question, which may be easily tried as before in the last, &c. Case 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 requisite to compose the Mixture, Note, This is usually called Alligation Partial. Queft. 4. How much Wheat at 5 s. the Bushel, must be mixed with 12 Bushels of Rye at 3 s. 6 d. a Bufhel; that the whole Mixture may be sold for 4 s. 4 d. the Bushel ? In this case you must set down all the particular Rates, with the Mean Rate, and find their Differences just as before ; without any regard had to the Quantity given. 10 Thus, Mean Rate 52 d.Wheat 60 d. U .Rye 42 d.) 18. r. As the Quantity found by the Differences of the same Name with the Quantity given : Is to the Quantity given :: Se 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 45. 3 d. the Gallon, must be mixed with 18 Garons of Canary at 6 s. 9 d. the Gallon; that the whole Mixture may be sold for 6's, the Gallon ? The The Terms being sec down, &c. as before, will stand Malaga 90 d. S 21 Thus, Mean Rate 72 d. 3 Sherry 6o d.? S 9 White 57 d. S218, (Canary 81 d. N 12 . (21 : 31 Gallons of Malaga. Then, as 12:18:: 318 : 27 Gallons of White. 19:13 ; Gallons of Sherry. That is, 31 į Gallons of Malaga, 27 of V[hite Wine, and 131 of Sherry, being mixed with 18 Gallons of Canary, will make the Mixture required. Malaga 90 512 Or thus, 723 Sherry 60 $ 18 Canary 81 U21 712 : 10 À the Malaga. 2 Then, 21:18:: 318 : 15 the Sherry. &c. (9: 7 if the White. ) Gallons. Pence. pio é at 90 d. r 925 Proof. J 15 i at god. Learn 925 | 7 at 51 d. 393 i li8 at 81 d.) (1458 Sum 51 & Value = 3702 11 the Mean Rate. Then 51 A) 37021 (72 d. = 6 s. Therefore the Quantities are as truly assigned 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; í As the Sum of all the Differences : Is to the Sum of all the Then Quantities given :: So is every particular Difference : Ç To it's particular Quantity. Queft. 6. Let it be required to mix Wheat at 5 s. the Bushel, with Rye at 3 s. 6 d. the Bushel ; so that the whole Quantity may be 27 Bushels, to be sold for 45. 4 d. a Bushel, what Quantity of each must be taken to make up the Mixture? Mean Mean Rate 52 { Ryes 422.318 18= their Sum. {the Quantities required. Sherry. 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 55. the Gallon, and White Wine at 4's. 3 d. the Gallon; so that the whole Mixture may be go Gallons ; to be fold for 6 s, the Gallon : How much of each fort will compose that Mixture ? Malaga 90 S 21 White 51 Sl18 Canary 8113 9 60 their Sum. Malaga. White Wine. Canary. Sherry 60 5 118 Canary 81 U21 - 60 their Sum. Malaga. 0 Sherry. Canary. ( 9:13 ) LWhite Wine. Either of these Ways do equally answer the Question, as may be easily tried by Alligation Medial. As before, &c. Note, The Work of these Proportions may be much fortened (especially when there are many Ingredients to be mixed) if you observe the fame Method as was proposed in the Rule of Fellowship, page 99, &c. I have made Use of the very fame Examples both in Alligation Medial, and Alternate, throughout the three Cafes ; being, as I presume, much better than if they had been different ones; because the Learner may (if he consider them a little) easily perceive, not only the Difference between the two Rules, but also wherein the |