30 240 gallons, at 75 cts. 5: 40 gallons, at 90 cts. Ans. 20: 160 gallons, at 125 cts. 2. How much gold, of 16, 20, and 24 carats fine, and how much alloy, must be mixed with 10 ounces, of 18 carats fine, that the composition may be 22 carats fine? Ans. 10 oz. of 16, 10 oz. of 20, 170 of 24 carats fine, and 10 oz. of alloy. 3. A grocer has 14 cwt. of sugar, that cost him $14 per cwt.; but being too dear to sell readily, he wishes to mix it with some at $10, some at $8, and some at $6 the cwt., and to make a composition that he can sell at $9 the cwt. and make 50 cents the cwt.; how much of each kind must he put with his 14 cwt.? Ans. 30 cwt. at $6. 83 cwt. at $8. 24 cwt. at $10. 4. A tobacconist mixed 20 lbs. of tobacco, worth 15 cts. a lb., with others, at 16 cts., 18 cts., and 22 cts. a lb.; how many lbs. of each, must he take, that the mixture may be worth 17 cts. Ans. 4 lbs. at 16 cts. 4 lbs. at 18 cts. 8 lbs. at 22 cts. CASE THIRD. Q. What is the third case in Alligation Alternate? A. It is when the prices 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 the desired compound. Q. What is the RULE in this case? A. Find the difference between the mean rate and the several prices, and place them as in case first. Then say, as the sum of all these differences: is to the given quantity :: so is each difference to its own particular quantity. EXAMPLES. 1. A person wishes to ship 2000 lbs. of cheese, that may average 9 cts. per lb. ; he has four sorts on hand, some at 12 cts., some at 10 cts., cts., and 6 cts. the lb.; how much of each sort will make up the required lot? 2. A goldsmith wishes to make a compound of 78 ounces, of 16 carats fine; and having gold of 14, 18, and 22 carats, how many ounces of each, and how many of alloy, will make the desired composition Ans. 6 oz. of 22 carats, 48 oz. of 18 carats, 18 oz. of 14 carats, and 6 ounces of alloy. 3. A grocer receives an order for 1000 pounds of sugar, at 8 cents per. lb.; but having none that he can sell at that price, he mixes different qualities; some at 12 cents, some at 9 cents, and some at 6 cents per pound; how much of each kind will make the mixture? Ans: 5555 lbs. at 6 cts., 2223 lbs. at 9 cts. and 222 lbs. at 12 cts. 4. I have 4 sorts, of wine; some at $33, some at $23, some at $14, some at 75 cents, of which I wish to make a mixture of 136 gallons, that I can sell at $2 a gallon; how much of each must I use? Ans. 51 gals. at 75 cents, 17 gals. at $1,25, 25 gals. at $2,50, and 42 gals. at $3,50 cts. POSITION. Q. What is Position, or what does it teach? A. It teaches, by taking false or supposed numbers, and proceeding with them according to the directions of the questions, to find the true ones required; and is divided into two parts, single, and double position. SINGLE POSITION. Q. What is Single Position, and what does it teach? A. Single position is when the results of the operation are proportional to their supposition. It teaches to solve those questions that require the multiplication or division of the number sought, by any proposed number; or to be increased or diminished by itself a certain number of times. Q. What is the RULE for single position? A. Take any convenient number and perform the same operation with it, as if it were the true one; then say, as the result arising from, this operation is to the given sum, in the question so is the supposed sum: to the true one required. EXAMPLES. 1. A.'s age is double that of B., and B.'s age is triple that of C., and the sum of all their ages is 140; what is the age of each ? Operation. Suppose C.'s age to be 25 years. Then C's age X3 B.'s=75 250 sum of their ages. Then as 250: 140 :: 25: 14-C.'s age. Then C.'s age being 14 years, 14x3=42 B.'s age. 42x2=84-A.'s age. 2. A person being asked how much money he had in his purse, replied, that if of the sum were multiplied by 7, and of the sum were added to the product, the amount would then be $292; how much had he in his purse? Ans. $60. 3. A boy being asked how many marbles he had, said that if,, and, were added to the number, he should then have 125; how many had he? Ans. 60. 4. A general, in a disastrous engagement, had of his army taken prisoners, one third of them killed and wounded, and 700 remaining unhurt; what number of men did he command? Ans. 4200 men. 5. A gentleman collected $78, to be distributed among a company of Irish emigrants, consisting of men, women, and children; to each man, he gave $6, to each woman he gave $4, and to each child, $2, there were twice as many women as men, and three times as many children as there were women; how many were there of each? Ans. 3 men, 6 women, and 18 children. 6. A. B. and C. bought a quantity of goods, which cost them $360"; they agreed to divide them so, that A., should have a certain portion, that B. should have more than A. and that C. should have 4 more than B., and that each should pay in that proportion; what proportion of the goods had each, and how much did each pay for his share? Ans. A. had, cost $90. B. 3, cost $120. C., cost $150. 7. What number is that, which being increased by its half, its third, and its fourth, the sum will be 160 ? Ans. 76,8. 8. A man having bated a flock of pigeons, set his net to catch them; the first day he caught one fourth of the flock, the second day he caught one third of the remaining part, the third day he caught one half of the residue, and the fourth day he caught 60, which comprised the whole flock; how many did the whole flock contain? Ans. 240. DOUBLE POSITION. Q. What is Double Position? A. It is when the number sought is increased or diminished by some given number, which is no known part of the number required, and includes those questions, whose results are not proportioned to their positions. Q. How do you solve questions in this rule? A. By supposing two false numbers, and proceeding with them as the question directs, and from their result the true number is found. Q. What is the RULE in double position? A. Take any convenient number, and proceed with it exactly as if it were the true number, and find the result; then find the difference between this result and the given sum in the question, and call it the first error. Next, suppose another convenient number, either greater or less than the first, and proceed with it as before, and call the difference between this second result and the given sum, the second error. Then multiply the first supposition into the second error, and the second supposition into the first error, and, if your errors are alike, divide the difference of these products, by the difference of the errors; but if the errors are unlike, divide the sum of the products, by the sum of the errors, and in either case the quotient will be the answer, or number sought. Q. When are the errors said to be alike or unlike? A. They are alike, when both the suppositions are either greater or less than the true number; they are unlike when one of the suppositions is greater, and the other is less than the true number required in the question. EXAMPLES. 1. A shrewd boy being asked how old he was, replied; 8 years ago, I was only one third as old as my father was at that time, but now, I am half as old as my father is at the present time; what was the age of the father and son Ans. The father's age, 32 years. Boy's, 16 years. 2. A boy, carrying some eggs to market, was asked how many he had to sell, to which he replied that he did not know exactly, but his father told him, if one dozen were added to the number, and that sum divided by 2, and this quotient multiplied by 4, and two dozen added to this product, and the whole of them sold for 12 cents a dozen, he would have $1,50 to carry home; how many eggs had he to sell? Ans. 4 dozen. 3. A drover, going to market with a drove of calves, sheep, and hogs, was met by an inquisitive fellow, who was very anxious to know how many animals he had in his drove, to whom he replied, that he should not tell him how many he had, but that he had made this calculation upon them; that his whole drove cost him $400; that of his drove were sheep; of the residue were hogs; and the remainder were calves; and, that if he could sell his sheep for $2 a head, his hogs for $31, and his calves for $5 a head, he should make $119 by his speculation; how many of each kind had he in his drove? Ans. 60 sheep, 54 hogs, and 36 calves. 4. A butcher bought at one time, 20 sheep, and 30 lambs, for which he gave $120; he afterwards bought 30 sheep, and 25 lambs, at the same price as before, which amounted to $140; what was the price paid for each sheep, and how much for each lamb? Ans. For each sheep, $3, and each lamb $2. 5. A gentleman has two horses, a black and a grey, and a chaise worth $150; if the black horse be put in the chaise, they will, together, be worth twice as much as the grey; but if the grey horse be put in the chaise, they will be worth three times as much as the black; what is the value of each horse? Ans. The black is worth $90, and the grey $120. 6. A boy, who had been stealing peaches, was caught by B. and to appease his anger, the boy gave him half he had, lacking 10; going farther, he met C. who took half he had left, lacking 4; after this he met D. who took from him half he had left, lacking 1; at length, getting safe away, he finds he has only 13 left; how many had he at first? Ans. 60. 7. There is a building, the posts of which are 14 feet in length; the width is equal to the height and half the length of the building, and the length is equal to the width and height added together; what are the dimensions of the building? Ans. 56 feet long, and 42 feet wide. 8. A man driving a flock of live geese to market, was met by a fellow, who thus accosted him: "Good morning, my friend, with your five hundred geese." "Ah!" said the man, "if you are a yankee, you have not guessed right this time. But I'll tell you what, if I had the flock of wild geese I saw in the pond, just back, I should then have 4350." "Indeed!" said the fellow, " why, how many did you see?" "I do n't know how many," said the man, "but I counted as many again as I have in my flock, my partner counted as many again as I did, and my son counted as many again as my partner, and yet we counted but just half the flock. Now, if you can tell how many geese I have in my flock, I will venture to give you How many had he? one. Ans. 150. |