ALLIGATION ALTERNATE SECT. III. 9. Is the method of finding what quantity of any number of simples, whose rates are given will compose a mixture of a given rate; it is, therefore, the reverse of Alligation Medial, and may be proved by it. RULE. 1. WRITE the prices of the simples, the least uppermost, &c. in a column under each other. 2. CONNECT with a continued line the price of each simple or ingredient, which is less than that of the compound, with one or any number of those that are greater than the compound, and each greater rate or price with one or any number of those that are less. 3. WRITE the difference between the mean rate or price and that of each of the simples, opposite to the rates with which they are connected. 4. THEN if only one difference stand against any rate it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity. NOTE. QUESTIONS in this rule admit of as many various answers as there are various ways of connecting the rates of the ingredients together. EXAMPLES. A GOLDSMITH would mix gold of 18 carats fine with some of 16, 19, 22 and 24 carats fine, so that the compound may be 20 carats fine; what quan tity of each must he take? OPERATION. oz. car. fine. PROOF. 16X4 64 2 18 18X2 36 19×2=38 22 2X1 22 22X3=66 24. 4 24×4-96 2. A DRUGGIST has several sorts of Tea. viz. one sort at 12s. per lb. a15)300(20 car fine, nother at 11s. a third at 9s. and a fourth at 8s. per lb. I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10s. per lb. NOTE. THESE Seven Answers arise from as many different Ways of linking the Rates of the Simples together. CASE 2. WHEN the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture are given to find the several quantities of the rest, in proportion to the given quantity; take the difference between each price and the mean rate as before. Then say, As the difference of that simple whose quantity is given, Is to the given quantity, So is the rest of the differences severally ; To the several quantities required. EXAMPLES. 1. How much wine, at 80 cents, at 88, and at 92 cents per gallon must be mixed with four gallons of wine at 75 cents per gallon, so that the mixture may be worth 86 cents per gallon? :{ gal. cts. 8:4 at 80 88 per gal. The answer: 17: 81/ 2. A MAN being determined to mix 10 bushels of wheat at 4s. per bushel, with rye at 38. with barley at 2s. and with oats at 1s. per bushel; I demand how much rye, barley, and oats must be mixed with the 10 bushels of wheat, that the whole may be sold at 28d. per bushel ? 1. Ans.. (B. 4. Ans.. 50 of Barley 12 2 of Oats (B. 10 of Rye 7. Ans.. [B. 2. Ans. 5. Ans. B. 50 of Barley 17 2 of Oats 50 of Rye CASE 3. WHEN the rates of the several ingredients, the quantity to be compounded, and the mean rate of the whole mixture are given to find how much of each sort will make up the quantity; find the differences between the mean rate, &c. as in case 1. Then, As the sum of the quantities, or differences, Is to the given quantity, or whole composition; So is the difference of each rate, To the required quantity of each rate. EXAMPLES. 1. How many gallons of water, of no value, must be mixed with brandy, at one dollar twenty cents per gallon so as to fill a vessel of 75 gallons, that may be afforded at 92 cents per gallon? 75 given quantity. 2. SUPPOSE I have 4 sorts of currants of 8d. 12d. 18d. and 22d. per lb. of which I would mix 120/b. and so much of each sort as to sell them at 16d. per lb, how much of each must Í take?' d 3. A GROCER has currants of 4d. 6d. 9d. and 11d. per lb. and he would make a mixture of 2406. so that it might be afforded at 8d. per lb, how much of each sort must he take? SUPPLEMENT to Alligation. 1. WHAT is Alligation? QUESTIONS. 2. Or how many kinds is Alligation ? 3. WHAT is Alligation MEDIAL? 4. WHAT is the rule for operating ? 5. WHAT is Alligation ALTERNATE ? 6. WHEN a number of ingredients of different prices are mixed together, how do we proceed to find the mean price of the compound or mixture ? 7. WHEN one of the ingredients is limited to a certain quantity, what is the method of procedure? 8. WHEN the whole composition is limited to a certain quantity, how do you proceed? 9. How is Alligation proved? EXERCISES. 1. A GROCER would mix three sorts of sugar together; one sort at 10d. per lb. another at 7d. and another | at 6d. how much of each sort must he take that the mixture may be sold for 8d. per lb ? 2. A GOLDSMITH has several sorts of gold; some of 24 carats fine, some of 22, and some of 18 carats fine, and he would have compounded of these sorts the quantity of 60 oz. of 20 carats fine; I demand how much of each Ans. 3lb. at 10d. 2 at 7d. and 2 at 6d. sort he must have? Ans. 12oz. 24 carats fine, 12 at 22 car- § 10. Position. POSITION is a rule which, by false or supposed numbers, taken at pleasure, discovers the true one required. It is of two kinds, SINGLE and DOUBLE Single Position. Is the working with one supposed number, as if it were the true one, to find the true number. RULE. 1. TAKE any number and perform the same operations with it as are described to be performed in the question. 2. THEN say; as the sum of the errrors is to the given sum, so is the sup posed number to the true one required. PROOF. ADD the the several parts of the sum together, and if it agree with the sum, it is right. EXAMPLES. 1. Two men, A and B, having found a bag of money, disputed who should have it; A said the half third, and one fourth of the money made 130 dollars, and if B could tell how much was in it, he should have it all, otherwise he should have nothing; I demand how much was in the bag ? |