EXAMPLES. 1. If an annuity of 125 dollars be forborne 4 years, what will be its amount, at compound interest? Ans. $546,81. (1,06)1× 125-125-32,8087,06)32,8087(546,81, Ans. 2. Find the amount of an annuity of 260 dollars, in arrears for 3 years, at 7 per cent. compound interest? Ans. $835,87+ [NOTE. Powers of the ratio can be found in Table 1, Art. 97, as the numbers in that table are nothing but powers of interest ratios; but it is unnecessary to multiply examples.] ALLIGATION. MEDIAL and alternate: a commingling, or throwing together. Alligation medial, is finding the medium or middle quality, or value, of certain amounts of given things put togther. Alligation alternate is finding the quality or value of several things, corresponding to a given medium; hence, in certain respects, one is the reverse of the other. Neither of them are of much practical value. But, as an improvement to the mind, and as a study of numbers, they form a good lesson. (ART. 147.) 1st. Alligation medial. From the definition of finding a medium, we recognize the following RULE. Find the value of each ingredient, and divide their sum by the number of ingredients. EXAMPLES. 1. A grocer mixed 10 pounds of sugar, at 8 cents, with 12 pounds at 9 cents, and 16 pounds at 11 cents; what was a pound of the mixture worth? Solution: Whole cost, 364 cents; divided by 38 lbs., gives 911 cts. answer. 2. If 120 bushels of wheat be bought at 80 cents per bushel, and 75 bushels at 86 cents a bushel, what is the average cost per bushel of the whole purchase? Ans. 82+cents. 3. An innkeeper mixed 13 gallons of water with 52 gallons of brandy, which cost him $1,25 per gallon: what is the value of 1 gallon of the mixture, and what his profit on the sale of the whole at 6 cents per gill? $65 profit. S$1 a gallon. Ans. This requires more (ART. 148.) Alligation alternate. explanation; and, for the purpose of explaining, let us take the following problem: A grocer has sugars, at 9 cents and 16 cents per pound; he wishes to make a mixture worth 11 cents; what portions of each sort shall he take? If he takes 1 pound of each, and sells it at 11 cents, it is evident he would lose 5 cents on the 16 cent sugar, and gain only 2 cents on the 9 cent sugar. On that sup position, he would lose more than he would gain; and to equalize the value, he must take more of the 9 cent sugar; as much more as 9 is nearer to 11, than 16 is nearer to 11. That is, take the reciprocal difference from the mean price. Thus, 11 9 5 162 [NOTE. If the pupil had studied Natural Philosophy, I would compare the principle of this process to balancing weights on the long and short arm of a lever: the smaller the difference the greater the quantity; the shorter the arm, the greater the weight, &c.] Το The difference between 9 and 11 is 2, put opposite 16, for the quantity; and the difference between 16 and 11 is 5, put opposite 9, for that quantity. The difference between the values and the mean value, are alternated. may therefore be designated alligation alternate. prove this result correct, we take alligation medial: 5 pounds at 9 cents, cost 45 cents, 2 pounds, at 16 cents, cost 32; the whole, 7 pounds, cost 77 cents, or 11 cents per pound, as required. It We may add one or two more qualities of sugar to the same question, and have it read thus: 1. A grocer has sugars at 9, 10, 14, and 16 cents per 9. pound, and wishes to make a mixture, worth 11 cents; how much of each quality shall he take? Take a balance between 9 and 16, as before; then between 10 and 14. The connecting lines show what terms are linked together. One term or ingredient may be taken to balance se 1114 16 5312 veral, when there is but one above or below the mean price, and several on the other side. To prove that the above result is correct, let us compute the whole cost and find the average. Thus, 5 pounds at 9 cents=45 cents, 3 at 10=30, 1 at 14=14, 2 at 16=32. Whole sum, 121 cents, divided by 11 pounds, gives 11 cents per pound. 2. A merchant would mix teas at 70, 85 and 100 cents per pound, and sell the mixture at 75 cents; what proportion of each must he take? pounds at 100, and 10 pounds at 70, will correspound to 5 pounds at 85. Hence, we must take 35 pounds at 70, to correspond to 5 of each of the others. Or, we must take the quantities in this proportion, not necessarily this quantity. For instance, 35 is to 5 as 7 to 1. Now if we take 7 pounds at 70, and one pound each at 85 and 100, the whole 9 pounds will be worth 75 cents per pound. Therefore, after making a required mixture, we can proportion the different ingredients so that the mixture shall contain any required amount of one of them. From the foregoing we draw the following RULE. Set the several ingredients in order, one under the other, and the mean on one side. Connect one less than the mean with any one greater; or, if the case requires it, one less with several greater, or vice versa. Place the difference of each and the mean rate against the ingredient with which it is connected. If only one difference stand against any rate, it will be the required quantity of that ingredient; but if there be more than one, their sum will be the quantity required of that ingredient. EXAMPLES. 1. A merchant has spices at 32 cents, 40 cents and 64 cents per pound. He wishes to mix 5 pounds of the first with the others, so that the compound may be worth 48 cents. How much of each must he use? Ans. 5 pounds of the second, and 7 pounds 8 ounces of the third. Make a mixture first, without any regard to the 5 lbs.; then if against the first stands 5 pounds, the problem is solved, if not, proportion the whole so as to make the first 5 pounds. 2. A farmer wishes to mix 14 bushels of rye worth 50 cents per bushel, with corn at 40 cents, and oats at 30 cents per bushel, so that the mixture may be worth 37 cents per bushel; what quantities must he take of each? Ans. 14 bushels of corn and 32 of oats. 3. A goldsmith has gold of 15, 17, 20 and 22 carats fine, and would melt together of all these sorts, so much as to make a mass of 40 ounces 18 carats fine. How much of each sort is required? Ans. 16 ounces of 15, 8 of 17, 4 of 20, and 12 of 22. 4. A man filled a wine hogshead containing 63 gallons with a mixture of wine worth 120 cents per gallon, with some worth 160 cents, and water worth nothing. The mixture was worth 130 cents per gallon; how much of each did he take? (ART. 149.) FROM time immemorial, formal rules have been given in arithmetic, under Position, to solve a certain class of rather complex proportional questions, more properly belonging to algebra. Such rules were limited in their application, and could cover no questions involving powers and roots, as powers and roots are not proportional to each other. For example, 16 and 64 are square numbers, and their roots are 4 and 8, or as 1 to 2; but the numbers themselves, 16 and 64, are to each other as 1 to 4, a different relation. 1. A man having a purse of money, being asked how much was in it, answered, The square root of it, added to the half of it, make 220 dollars; how much was in the purse? It is evident, that this question must be excluded from proportional operation; for, unless we first suppose the right number, the result of the supposition will not be to the given result as the supposed number to the true number; and when this proportion fails, supposition, that is, position fails. We prefer to leave the following questions without rules other than general analysis and proportion. The explanations following some one or two will be sufficient. EXAMPLES. 1. One-half, one-third and one-fourth of a certain number, added together, make 130; what is the number? Ans. 120. We take the position, that the required number is a whole. Then, 1+1+1=13. Then, by proportion, 13: 130 :: 12: Ans. 2. A post is in the earth, in the water, and 13 feet above the water; what is the length of the post? Ans. 35 feet. Add and; not and of the number 1, but and of the whole post. These parts, added together, make ; the remaining must be 13 feet. Then, by pro 13 portion, 13 : 13 :: 35 |