CASE II. To find the quantity to be used of each article, when the average cost and the cost of each article are given. 1. A grocer wishes to mix sugar at 10 ct. a pound with some at 16 ct. a pound, so that a pound of the mixture shall be worth 11 ct.; how much of each must he take? 11 {10] f 10,5 pounds of 10-cent sugar. ANALYSIS.-It is evident that if the grocer puts in 1 pound of the 16cent sugar, and sells it for 11 cents, he loses cents, and as on each pound of 10-cent sugar he gains 1 cent, it will be necessary to mix 5 pounds of the 10-cent sugar in order that the gain on it shall exactly balance the loss on 1 pound of 16-cent sugar. He must therefore use 5 pounds of the 10-cent sugar to 1 pound of the 16-cent sugar. Note.—Analysis is a simple and philosophical method of solving examples in Alligation, and the pupils should be encouraged to solve all such problems by this method. RULE. Write the average cost by itself, and the cost of each article in a column on the right. Link each value that is less than the average cost with one that is greater. Place the difference between a less value and the average cost, opposite the greater value with which such less value is linked, and the difference between a greater value and the average cost, opposite the less value with which it is linked. If there is only one difference opposite to any value, it will be the required quantity of the article of that value; but if there should be two or more differences, their sum will express the required quantity. 2. What number of barrels of flour worth $7, $8, and $9 per barrel must be sold, to realize $8.50 as an average price per barrel? Note. Following the directions contained in the rule, we find that barrel of $7 flour, barrel of $8 flour, and 2 barrels of $9 flour will fulfil the conditions of the question. If we wish for any greater number of barrels, we can multiply each of the three quantities by the same number and still preserve the proportion. Thus, multiplying by 10, we shall have 5 barrels of $7, 5 of $8, and 20 of $9 flour. 3. In what quantities must teas be mixed, worth 75 ct., $1.00, $1.25, and $1.50 per pound, respectively, in order that a pound of the mixture shall be worth 85 cents? 4. How shall three grades of coffee be mixed, the first costing 20 ct. per pound, the second 24 ct., and the third 40 ct. so that a pound of the mixture can be sold for 30 ct. and a profit of 25% be made on the transaction? 5. I have apples worth 25 ct. a dozen, oranges worth 50 ct. a dozen, and lemons worth 40 ct. a dozen; how many dozen of each must I sell that the average price may be 45 cents? 6. Bought four kinds of cloth, at 75 ct., $1.25, $2, and $3 per yard; how many yards did I buy of each, if the average cost is $2.50 per yard? 7. How much water must be mixed with milk worth 10 ct. a quart, that it may be sold at 8 ct. a quart without loss to the seller? 8. How shall three qualities of gold, 12 carats, 20 carats, and 22 carats fine, respectively, be mixed so that the mixture shall be 18 carats in fineness? CASE III. When one of the articles is limited in quantity. 1. How shall 20 pounds of coffee, worth 20 ct. a pound, be mixed with coffee worth 24 ct. and coffee worth 40 ct. a pound, so that the mixture shall be worth 30 ct. a pound? ANALYSIS.-We find by Case II, without regard to the quantity of the 20-cent coffee, that the proportional parts are 10 lb. of 20cent, 10 lb. of 24-cent, and 16 lb. of 40-cent coffee. But as we have 20 lb. of 20-cent coffee we must take 18, or 2 times as much of each of the other kinds, or 20 lb. of 24-cent and 32 lb. of 40cent coffee. RULE. Find the proportional quantities of each article by Case II. Divide the given quantity by the proportional quantity of that article, and multiply the remaining proportional quantities by the quotient. 2. A wine merchant wishes to mix three grades of wine, worth $1, $2, and $3 per gallon, so as to sell the mixture at $2.75 per gal. Having but 20 gallons of the first, how much will he require of each of the others? 3. How much water must be mixed with 20 gallons of alcohol, worth $2 per gallon, so that a gallon of the mixture can be sold for $1.50? 4. Bought turkeys at $1 apiece, geese at 75 cents, ducks at 50 cents, and 15 chickens, at 30 cents each; how many were there of each, if the average cost was 60 cents? 5. How much sugar worth 12 cents a pound must be mixed with 10 pounds valued at 10 cents a pound, so that the mixture shall be worth 10 cents per pound? CASE IV. When two or more of the articles are limited in quantity. 1. How much gold, 10, 12, and 14 carats fine, must be mixed with 2 ounces of 20 carats, and 3 ounces of 22 carats fine, in order that the mixture shall be 18 carats fine? ANALYSIS.-We first find by Case I the average fineness of the two quantities given, which is 21 carats for the 5 ounces; then by Case II we find the proportional quantities required, to be 31 ounces each of the 10, 12, and 14 carats to 18 ounces of the mixture of 20 and 22 carats; but as we have only ounces of the mixture we will require (Case III) of 3 ounces= of the 10, 12, and 14 carats. oz. each RULE. Find the average value of articles that are limited in quantity, by Case I; the proportional quantities required, by Case II; and the amount of each article not limited in quantity, by Case III. 2. A grocer desires to mix coffee worth 16 cents a pound, with 15 pounds of an article worth 25 cents per pound, and 15 pounds worth 35 cents per pound, so that the mixture can be sold for 18 cents per pound, without loss; how many pounds of the first will there be in the mixture? 3. How much gold 15 carats fine must be added to 14 ounces 20 carats fine, and 2 ounces 21 carats fine, so that the mixture shall be 16 carats fine? 4. Bought flour at $4 and at $6 per barrel, and have on hand 20 barrels worth $5, and 20 barrels worth $10 per barrel; how many barrels did I buy if the average cost of all is $7? 5. A merchant has cloth worth $2 a yard, and cloth worth $4 a yard, and buys 50 yards at $5 a yard, and 100 yards at $3.50 a yard; how much has he of each of the first two, if the average cost of all is $3? CASE V. When the whole compound is limited to a particular quantity. 1. A grocer wishes to mix tea at 50 cents, 75 cents, and 80 cents a pound; how many pounds of each must there be in 100 pounds of the mixture, the average price being 60 cents a pound? |