RULE.

Divide the sum of the values of all the ingredients by the sum of the ingredients.

Examples.

1. A wine-merchant mixed several sorts of wine, viz: 32 gallons, at 40 cents per gallon; 15 gallons, at 60 cents per gallon; 45 gallons, at 48 cents per gallon; and 8 gallons, at 85 cents per gallon. What is the value of a gallon of the mixture?

32 gallons, at 40 cents=$12.80

45 66

6 60 66 = 9.00

"48 66

21.60

8 66 " 85 66 = 6.80

100 gallons of mixture =$50.20

Therefore, one gallon of the mixture is worth $50.20÷100 =$0.502=50 cents and 2 mills.

2. A farmer mixes together 7 bushels of rye, worth 72 cts. per bushel; 15 bushels of corn, worth 60 cts. per bushel; and 12 bushels of wheat, worth $1.20 per bushel. What is the value of a bushel of the mixture?

Ans. $0.83.

3. A goldsmith melts together 11 ounces of gold, 23 carats fine; 8 ounces, 21 carats fine; 10 ounces of pure gold, and 2 pounds of alloy. How many carats fine is the mixture?

Ans. 123.

It will be understood that a carat is a 24th part. Thus, 21 carats fine is the same as

23 carats fine is 22 pure metal.,

pure metal`; in the same way,

4. On a certain day, the mercury in the thermometer was observed to stand 2 hours at 62 degrees, 4 hours at 70 de

grees, 5 hours at 72

hour at 75 degrees. 15 hours?

degrees, 3 hours at 59 degrees, and 1 What was the mean temperature for the

Ans. 671 degrees.

What is

5. Suppose a ship sails at the rate of 5 knots for 3 hours, at 7 knots for 5 hours, and 8 knots for 4 hours. her rate of sailing during the 12 hours?

6. A grocer mixes 30 pounds of sugar, worth 10 cents per pound; 40 pounds, worth 10 cents; 24 pounds, worth 11 cents per pound; and 60 pounds, worth 13 cents per pound. What is a pound of the mixture worth?

ALLIGATION ALTERNATE.

87. ALLIGATION ALTERNATE is the reverse of Alligation Medial; that is, it teaches the method of finding the ingredients, when their rates are given, so that the compound shall have a given value.

Suppose we wished to mix teas, which are worth 4 and 6 shillings per pound, so that the mixture may be worth 5 shillings per pound, it is obvious that we must take equal quanti. ties of each; since the price of the one, is as much less than the mean price, as the other is greater.

Again, suppose we wish to mix teas, which are worth 4 and 7 shillings per pound, so that the mixture may be worth 5 shillings. In this case the 7 shilling tea is 2 shillings above the average price, whilst the 4 shilling tea is but 1 shilling below; it will be necessary to use twice as much of the 4 shilling tea as of the 7 shilling tea; and in all cases it is obvious that the quantities to be used will be in the inverse ratio to the differences between their prices and the mean price. When

there are more than two simples they may be compared together in couplets, one term of which must exceed the aver age price, whilst the other must be less.

CASE I.

The rates of the several ingredients being given, to make a compound of a fixed rate.

From what has been said above, we draw the following

RULE.

I. Write the rates of the simples in a line under each other, then connect each rate of the ingredients, which is less than the rate of the compound, with one or more rates greater than the rate of the compound; connect in the same way, each rate which is greater than the rate of the compound, with one or more rates which are less.

II. Write the difference between each rate of the ingredients, and the compound rate, opposite the rate of the ingredient with which it is connected. If only one difference stand against any rate, it will be the required quantity of the ingredient of that rate; but, if there be several, their sum will be the quantity required.

Examples.

1. How much sugar at 5, 6, and 10 cents per pound, must be mixed together, so that a pound of the mixture may be worth 8 cents?

Therefore, if we take 2 pounds at 5 cents, 2 pounds at 6 cents, and 5 pounds at 10 cents, we shall satisfy the condi

&

tions of the question. It is obvious that any other number of pounds which are to each other as the numbers 2, 2, and 5, will satisfy the question equally well; so that in Alligation the number of solutions are indefinite; all that we can do is to find the ratios of the quantities required.

NOTE.-In many cases the ingredients will admit of being connected in several ways, and then we shall obtain as many sets of ratios as there are methods of connecting them.

2. How many pounds of raisins at 4, 6, 8, and 10 cents per pound must be mixed, so that a pound of the compound may be worth 7 cents?

In this question the terms may be connected in seven distinct ways; therefore we shall obtain seven sets of ratios, as follows:

NOTE. The two remaining methods of connecting the terms, have been omitted on account of the difficulty in arranging the type in imitation of the operation.

In one of these cases, the 4 is connected with the 8 and 10; the 6 with 10, and the ratios are found to be 4, 3, 3, and 4. In the other case the 4 is connected with the 8 and 10; the 6 with 8, and 10, and the ratios are all equal.

3. How much wine, at 72 cents per gallon, and 48 cents

per gallon, must be mixed together, that the composition may be worth 60 cents per gallon?

Ans. An equal quantity of each.

4. Suppose ten pounds of pure gold, when immersed in water, to displace 4 pounds of water; 10 pounds of pure silver to displace 7 pounds of water; and 10 pounds of an alloy of gold and silver to displace 6 pounds of water. What are the proportions of gold and silver in the compound?

Ans. Twice as much silver as gold.

5. How many gallons of wine and water must be mixed together, so that the mixture shall be worth 60 cents per gallon, the water being considered of no value, and the wine with which it is mixed being worth 90 cents per gallon?

Ans. 2 gallons of wine to 1 of water.

6. Having gold of 12, 16, 17, and 22 carats fine, what proportion of each kind must I take, to make a compound of 18 carats fine?

7. It is required to mix different sorts of grain, at 56, 62, and 75 cents per bushel, so that the mixture may be worth 60 cents per bushel. How much of each kind must be taken?

8. Hiero, king of Sicily, ordered a crown to be made containing 63oz. of pure gold; but suspecting that the workmen had debased it by using part silver, he recommended the detection of the fraud to the famous Archimedes, who putting it into water found that it displaced 8.2245 cubic inches of water. He next found that a cubic inch of gold weighed 10.36 ounces, and a cubic inch of silver weighed 5.85 ounces; then, from this data, he calculated the proportions of gold and silver of which the crown was composed. What must have been his

result?

Ans. 2.5447 oz. of gold to every 2.1434 oz. of silver.