worth $1.10 per bushel, desires to form a mixture worth $.50 per bushel, which shall contain equal parts of corn and wheat; in what proportion shall the ingredients be taken?

couplets, or any multiples of both, together. Multiplying the terms in column 4 by 2, we obtain the terms in column 5; and adding the terms in columns 3 and 5, we obtain the terms in column 6; that is, the farmer takes 7 bushels of oats to 2 of corn and 2 of wheat, which is the required proportion. Hence the following

RULE. I. Compare the given prices, and obtain the proportional terms by couplets, as in Case II.

II. Reduce the couplets to higher or lower terms, as may be required; then select the columns at pleasure, and combine them by adding the terms in the same horizontal line, till a set of proportional terms is obtained, answering the required conditions.

EXAMPLES FOR PRACTICE.

1. A grocer has four kinds of molasses, worth $.25, $.50, $.62, and $.70 per gallon, respectively; in what proportions may he mix the four kinds, to obtain a compound worth $.58 per gallon, using equal parts of the first two kinds? Ans. 4, 4, 8 and 11.

2. In what proportions thay we take sugars at 7 cts., 8 cts., 13 cts., and 15 cts., to form a compound worth 10 cts. per pound, using equal parts of the first three kinds? Ans. 5, 5, 5 and 2.

3 A miller has oats at 30 cts., corn at 50 cts., and wheat at 100 cts. per bushel. He desires to form two mixtures, each worth 70 cts. per bushel. In the first he would have equal parts of oats and corn, and in the second, equal parts of corn and wheat; what must be the proportional terms for each mixture?

CASE IV.

636. When the quantity of one of the simples is limited.

1. A miller has oats worth $.28, corn worth $.44, and barley worth $.99 per bushel. He wishes to form a mixture worth $.58 per bushel, and containing 100 bushels of corn. How many bushels of oats and barley may he take?

ANALYSIS. By Case II, we find the proportional quantities

to be 7 bushels of oats to 5 of corn and

8 of barley. But as 100 bushels of corn, instead of 5, are required, we must take 19° 20 times each of the other ingredients, in order that the gain and loss may be equal; and we shall therefore have 7 x 20 140 bushels o. oats, and 8 × 20 = 160 bushels of barley. Hence the following

RULE. Find the proportional quantities by Case II or Case III. Divide the given quantity by the proportional quantity of this ingredient, and multiply each of the other proportional quantities by the quotient thus obtained.

EXAMPLES FOR PRACTICE.

1. A dairyman bought 10 cows at $20 a head; how many must he buy at $16, $18, and $24 a head, so that the whole may cost him an average price of $22 a head?

Ans. 10 at $16, 10 at $18, and 60 at $24. 2. Bought 12 yards of cloth for $15; how many yards must I buy at $13, and $3 a yard, that the average price of the whole may be $1? Ans. 12 yards at $13 and 16 yards at $4. 3. How much water will dilute 9 gal. 2 qt. 1 pt. of alcohol 96 per cent. strong to 84 per cent.? Ans. 1 gal. 1 qt. 1 pt.

4. A grocer mixed teas worth $.30, $.55, and $.70 per pound respectively, forming a mixture worth $.45 per pound, having equal parts of the first two kinds, and 12 lbs. of the third kind; how many pounds of each of the first two kinds did he take?

CASE V.

637. When the quantities of two or more of the ingredients are limited.

1. How many bushels of rye at $1.08, and of wheat at $1.44, must be mixed with 18 bushels of oats at $.48, 8 bushels of corn at $.52, and 4 bushels of barley at $.85, that the mixture may be worth $.84 per bushel ?

ANALYSIS. Of the given quantities there are 18+ 8430 bushels, whose mean or average price we find by Case I to be $.54 We are therefore required to mix 30 bushels of grain worth $.54 per bushel, with rye at $1.08, and wheat at $1.44, to make a compound worth $.84 per bushel. Proceeding as in Case IV, we

find there will be required 25 bushels of rye, and 5 bushels of wheat. Hence the following

RULE. Consider those ingredients whose quantities and prices are given as forming a mixture, and find their mean price by Case I; then consider this mixture as a single ingredient whose quantity and price are known, and find the quantities of the other ingredients by Case IV.

EXAMPLES FOR PRACTICE.

1. A gentleman bought 7 yards of cloth @ $2 20, and 7 yards @ $2; how much must he buy @ $1.60, and @ $1.75 that the average price of the whole may be $1.80?

2. How much wine, at $1.75 a gallon, must be added to 60 gallons at $1.14, and 30 gallons at $1.26 a gallon, so that the mixture may be worth $1.57 a gallon? Ans. 195 gallons.

3. A farmer has 40 bushels of wheat worth $2 a bushel, and 70 bushels of corn worth $ a bushel. How many oats worth $1 a bushel must he mix with the wheat and corn, to make the mixture worth $1 a bushel? Ans. 6 bushels.

CASE VI.

638. When the quantity of the whole compound is limited.

1. A tradesman has three kinds of tea rated at $.30, $.45, and $.60 per pound, respectively; what quantities of each should he take to form a mixture of 72 pounds, worth $.40 per pound?

would form a mixture of 6 pounds. And since the required mixture is 7=12 times 6 pounds, we multiply each of the proportional terms by 12, and obtain for the required quantities, 36 lb. at $.30, 24 lb. at $.45, and 12 lb. at $.60. Hence the following

RULE. Find the proportional numbers as in Case II or Case III. Divide the given quantity by the sum of the proportional quantities, and multiply each of the proportional quantities by the quotient thus obtained.

EXAMPLES FOR PRACTICE.

1. A grocer has coffee worth 8 cts., 16 cts., and 24 cts. per pound respectively; how much of each kind must he use, to fill a cask holding 240 lb, that shall be worth 20 cts. a pound?

Ans. 40 lb. at 8 cts, 40 lb. at 16 cts., and 160 lb. at 24 cts.

2. A man bought calves, sheep, and lambs, 154 in all, for $154. He paid $3 for each calf, $13 for each sheep, and $1 for each lamb; how many did he buy of each kind?

Ans. 14 calves, 42 sheep, and 98 lambs. 3. A man paid $165 to 55 laborers, consisting of men, women, and boys; to the men he paid $5 a week, to the women $1 a week, ad to the boys $ a week; how many were there of each?

Ans. 30 men, 5 women, and 20 boys.

INVOLUTION.

639. A Power is the product arising from multiplying a number by itself, or repeating it any number of times as a factor. 640. Involution of the process of raising a number to a given

power.

641. The Square of a number is its second power.

642. The Cube of a number is its third power.

643. In the process of involution, we observe,

L. That the exponent of any power is equal to the number of times the root has been taken as a factor in continued multiplication. Hence

II. The product of any two or more powers of the same number is the power denoted by the sum of their exponents, and

III. If any power of a number be raised to any given power, the result will be that power of the number denoted by the product of the exponents.

1. What is the 5th power of 6?

tiplication; the final product, 7776, is the power required, (I). Or, we may first form the 2d and 3d powers; then the product of these two powers will be the 5th power required, (II).

2. What is the 6th power of 12?

122

OPERATION.

144

144 2985984, Ans.

ANALYSIS. We find the cube of the second power, which must be the 6th power, (III).

644. fence for the involution of numbers we have the fol