EXAMPLES. 1. If 19 bushels of rye, at 30 cents per bushel; 40 of corn, at 20 cents; and 12 of oats, at 15 cents, be mixed together, what is a bushel of this mixture worth? 19 × 30 = 570 40 X 20 = 800 12 x 15 = 180 71 71)1550(215 cents. Answer. 2. If 4 ounces of silver, worth 60 cents the ounce, be melted with 8 ounces, at 48 cents, what is 1 ounce of the mixture worth? 3. A wine merchant mixes 12 gallons of wine, at 58 cents the gallon, with 24 gallons, at 66 cents, and 16 at 75 cents: what is a gallon of this mixture worth? 4. A goldsmith melted together 8 ounces of gold 22 carats fine, 1 lb. 8 ozs., 21 carats fine, and 10 ozs., 18 carats fine what is the quality of the composition? 5. A refiner melted 5 lbs. of silver bullion 8 ozs. fine, with 10 lbs. of 7 ozs., and 15 lbs., 6 ozs. fine: of what fineness is 1 lb. of this mass? 6. A composition is made of 18 lbs. of tea, at 66 cents per lb., with 20 lbs. at 69 cents, 24 lbs. at 75 cents, and 16 lbs. at 78 cents per lb.: what is the worth of 3 lbs. of this mixture? 7. A vintner compounds a pipe of wine, of 36 galls. at $1.44 per gall., 40 galls. at $1.56 per gall., and 50 galls. at $1.68 per gall.: what will a gallon of this mixture be worth? § 179. CASE 2. Alligation alternate. When the rates of several things are given, to find what quantity of each must be taken to make a mixture of a certain mean value. RULE. 1. Place the rates of the ingredients under each other, and place the mean rate on the left hand of them. 2. Link the several rates together, so that one greater than the mean rate may be joined to one that is less. 3. Take the difference between each price and the mean rate, and place it opposite to the rate to which it is linked. 4. If only one difference stand against any rate, that difference will be the answer; but if more than one, their sum will be the answer. Proof. By alligation medial. EXAMPLES. lb.: 1. A grocer would mix sugar, at 10 cents, 9 cents, 7 cents, and 6 cents per lb., to make a mixture worth 8 cents per how much of each sort must he take? 2. A tobacconist would mix tobacco at 42 cents, 36 cents, 24 cents, and 18 cents per lb.: what quantity of each must he take to make a mixture worth 30 cents per lb. ? 3. A maltster has several sorts of malt, at. 48 cents, 60 cents, 72 cents, and 78 cents per bushel : how much of each sort must be taken to make a mixture worth 66 cents per bushel ? 4. What quantity of raisins at 71, 61, and 51 cents per pound, must be mixed together to sell at 6 cents per pound? 5. How much wheat at 60 cents, barley at 48 cents, and oats at 36 cents, will make a mixture worth 42 cents per bushel? 6. A brewer had ale at 16 cents, 12 cents, and 8 cents per gallon how much of each sort must he take to sell at 10 cents per gallon? 7. A tea dealer has several sorts of tea, at 55 cents, 45 cents, 40 cents, and 35 cents per pound: how much of each sort must be used that the whole quantity may be afforded at 50 cents per pound? lb. 5+10+15=30 at 55 pr. lb. 30x55=16.50 Proof. lb. cts. $ 8. How many ounces of gold, of 22, 18, 17 and 16, carats fine, must be mixed, so that the composition may be 20 carats fine? 9. How much sugar, at 7 cents, 8 cents, 9 cents, and 16 cents per pound, must be mixed together to make a mixture that may be sold at 10 cents per pound? § 180. CASE 3. Alligation partial is when one of its ingredients is limited to a certain quantity. RULE. 1st. Take the difference between each price and the mean rates as before. 2d. State, as the difference of that commodity whose quantity is given is to the rest of the differences severally, so is the quantity given to the several quantities required. EXAMPLES. 1. A farmer would mix 54 bushels of wheat at 90 cents per bushel, with rye at 54 cents, and barley at 63 cents per bushel, to make a mixture worth 72 cents per bushel. 2. A distiller would mix 30 gallons of French brandy at $2.88 per gallon, with English at $1.44, and spirits at 96 cents per gallon: what quantity of each must be taken to be afforded at $1.92 per gallon? 3. A grocer mixes 24 lb. of fine tea at 90 cents per lb. with others at 65 cents and 60 cents per lb. to make a mixture worth 75 cents per lb.: what quantity of each does he take? 4. How much wine at $1.26, $1.50, and $2.16 per gallon, must be mixed with 18 gallons at $1.98 per gallon, to make a composition worth $1.80 per gallon? 5. A grocer would mix 56 lbs. of tea at 36 cents per lb. with others at 45 cents, 51 cents, and 54 cents, to make a composition worth 48 cents per lb. : how much of each must he take? 6. A mealman mixes 60 bushels of flour at $1.26, with others at $1.08 and 96 cents, to make a mixture worth $1.14 per bushel: what quantity of each does he take? § 181. CASE 4. Alligation total is when the whole of the ingredients are limited to a certain quantity. RULE. 1st. Take the difference between each price and the mean rate, as before. 2d. State, as the sum of the differences is to each particular difference, so is the quantity given to the quantity required. EXAMPLES. 1. A grocer has sugar at 12 cents, 10 cents, and 8 cents per lb., and he would make a composition of 54 lbs., to sell for 9 cents per lb.: how much of each must he take? 2. A druggist who has drugs at 96 cents, 60 cents, and 48 cents per lb., would make a composition of 112 lb. worth 72 cents per lb.: what quantity of each must he take? 3. A goldsmith has several sorts of gold, some of 24 carats fine, some 22, and some 18 carats fine, with which he would make make a compound of 30 ounces of 20 carats fine how much of each sort must he take? 4. A person has rice at 12 cents, 7 cents, 6 cents, and 5 cents per lb., with which he makes a composition of a quarter of a cwt., worth 8 cents per lb.: what quantity of each does he take? lbs. oz. |1+2+3=6|As 18:6:28:9 5 at 12 cents. 4 18:4:28:6 35 66 7 66 18:4:28:6 35 66 6 66 5. A wine merchant has four sorts of wine, viz. Canary at 70 cents per gallon, Malaga at 65 cents, Rhenish at 55 cents, and Oporto at 50 cents per gallon; and he is desirous of making a composition of a pipe (126 gallons) to sell for 60 cents per gallon: the quantity of each is required? 6. I have teas at 48 cents, 60 cents, 84 cents, and $1.08 per lb., and I would make a mixture of a cwt. (56 lbs.) to sell at 72 cents per lb. ; what quantity of each will be required? POSITION. $182. Position is the method of performing such questions as cannot be resolved by the common direct rules, and is of two kinds, called single and double. SINGLE POSITION. § 183. Single position teaches to resolve those questions whose results are proportioned to their suppositions. RULE. 1. Take any number, and perform the same operations as are directed to be performed in the question. 2. Then say, as the result of the operation is to the result in the question, so is the supposed number to the number required. Any supposed number will produce the true answer; but for convenience in working, those numbers are to be preferred, from which all the parts can be taken without remainders. Some, however, recommend the number one to be made the constant supposition? Only those questions belong to single position whose parts are certain proportions of the suppositions, or of some power or root of their suppositions. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, if I had as many more, half as many, and a quarter as many, I should have 330: how many had he? 2 A person, after spending and of his $60 left: what had he at first? money, had |