(11.) A goldsmith has gold of 16, of 18, of 23, and of 24 carats fine; what part of each must be taken so that the mixture shall be 21 carats fine? Ans. 3 of 16; 2 of 18; 3 of 23; and 5 of 24. (12.) What proportions of coffee at 16, 20, and 28¢ per lb. must be mixed together so that the compound shall be worth 24 per lb.? Ans. In the proportion of 4lb. at 164; 4lb. at 204; and 12lb. at 28¢. (13.) What portion of brandy at 14s. per gallon, of old Madeira at 24s. per gallon, of new Madeira at 21s. per gallon, and of brandy at 10s. per gallon, must be mixed together so that the mixture shall be worth 18s. per gallon? Ans. 6 gal. at 10s.; 3 at 14s.; 4 at 21s.; and 8 at 24s. 14.) A grocer having four sorts of tea worth 5s., 6s., 8s., and 9s. per lb. wishes a mixture of 87lb. worth 7s. per lb.; how much of each sort must be taken? Ans. 291b. at 5s.; 144lb. at 6s.; 14 at 8s.; and 291b. at 9s. (15.) A vintner has 4 sorts of wine; white wine at 4s. per gal., Flemish at 6s. per gal., Malaga at 8s. per gal., and Canary at 10s. per. gal.; he would make a mixture of 60 gallons to be worth 5s. per gallon; what quantity must be taken of each ? Ans. 45 gal. of white wine; 5 gal. of Flemish; 5 gal. of Malaga; and 5 gal. of Canary. (16.) A silversmith has 4 sorts of gold; of 24, of 22, of 20, and of 15 carats fine; he would make a mixture of 42oz. of 17 carats fine; how much must be taken of each sort? Ans. 4 of 24; 4 of 22; 4 of 20; and 30 of 15 carats fine. (17.) How much wine at 5s., at 5s. 6d., and 6s. a gallon must be mixed with 4 gallons at 4s. a gallon so that the mixture shall be worth 5s. 4d. per gallon? Ans. 1 gal. at 5s.; 2 at 5s. 6d.; and 8 at 6s. (18.) A farmer would mix 14 bushels of wheat at $1,25 a bushel, with rye at $0,72, barley at $0,48, and oats at $0,36; how much must be taken of each sort to make the mixture worth $0,64 per bushel? Ans. 14 bu. of wheat; 8 bu. of rye; 4 bu. of barley; 28 hu. of oats. (19.) There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at 18d. per bushel; how much is taken of each sort when the mixture is worth 3s. 6d.? Ans. 96bu. of wheat; 12bu. of rye; 12bu. of barley; and 12bu. of oats. (20.) A distiller would mix 40 gal. of French brandy at 12s. per gallon, with English at 7s. and spirits at 4s. per gallon; what quantity must be taken of each sort, that the mixture may be afforded at 8s. per gallon? Ans. 40 gal. French; 32 gal. English; and 32 gal. of spirits. (21.) If 8 bushels of rye at 50 cents a bushel, be mixed with 12 bushels of corn worth 65 cents a bushel, and 6 bushels of oats at 30 cents a bushel; what is a bushel of the mixture worth? Ans. $0,523. (22.) Suppose that a number of men work at a certain work during a month, as follows, viz.: 6 men work 15 days each; 4 men work 19 days each; 12 men work 20 days each; and 10 men work 26 days each, during that time; on how many days' work, on an average, can one calculate for each man, in a month? Ans. 2013 days. (23.) A goldsmith having gold 15 carats fine, 19 carats, 21 carats, and 24 carats, wishes to make a mixture 20 carats fine; how much of each must he take? Ans. 440 carats. (24.) If a grocer have sugars worth 11 cents, 13 cents, 14 cents, 15 cents, and 16 cents a pound; in what proportions must he mix them, in order that the mixture may be worth 12 cents per pound? Ans. 10lb. at 114, and 1lb. of each of the others. (25.) What quantity of rye at 48 cents, of corn at 36 cts., and of barley at 30 cents a bushel, being mixed with 10 bushels of wheat worth 70 cents a bushel, will form a mixture worth 38 cents a bushel ? 2 bushels of rye. Ans. 12 do. corn. barley. (26.) A farmer mixed 15 bushels of rye, at 64 cents a bushel; 18 bushels of corn, at 55 cents a bushels; and 21 bushels of oats at 28 cents a bushel; what is a bushel of the mixture worth? Ans. $0,47, PART V. POWERS AND ROOTS; OR, INVOLUTION AND EVOLUTION. I. INVOLUTION. § 154. 1. WHEN two numbers which are equal are multiplied together, or a number is multiplied by itself, the product is surface (§114. 5); and is called a square, because it corresponds to what would be produced in nature by laying off the quantity which these numbers represent in any unit of lineal measure, in two directions perpendicular to each other, and completing the figure by two equal lines drawn perpendicular at the end of them.. Taking A B=4ft., and A C=4ft., and then drawing B D and C D perpendicular to A B and A C, and the square A B C D is the result; repD resenting the square of 4ft.; that is, 4ft. x4ft.= 16ft.; and contains 16 square feet. C 2. The product of any two numbers may be represented in the same way, by two lines perpendicular to each other, of given dimensions, and completing the figure which they indicate. е Taking A B=3ft., and A. C=6ft., and completing the figure, as above, by drawing B D and C D, and the rectangular figure A B C D is the result; representing the product of 3ft.×6ft.=18ft.; and contains 18 square feet. 3. It is evident then that when we have a surface of either of the kinds now considered and one of the sides given, the other side may be obtained by division, for in each case we have a product and one factor given to obtain the other factor (§ 36. 4). In the former case, indeed, the giving of one side gives the other, the two sides being equal, so that division to obtain it is not necessary. 4. But, when the figure is a square and the surface only is given, and the sides are required, a peculiar mode of operation is resorted to, which is called Extraction of the Square Root. § 155. 1. The multiplication of a number into itself, is called Involution; the result or product of such multiplication is called a power; and the number itself, a root. 2. The number of times a root is embraced in a product or power, numbers that power. Thus, if a number be multiplied by itself, the product is called the second power, or square; as 4×4=16, the second power or square of 4. If this power be again multiplied by the root, the product is called the third power, or cube; as 4x4x4=64, the third power, or cube of 4. If the third power be multiplied by the root, the product is the fourth power, or bi-quadrate; and so on. 3. In Arithmetic, the power to which a number is required to be raised is denoted by a figure written at the right, and a little above the number, which is called the index, or exponent of that power. Thus, if we involve 2, we have, 2=2, the root. 2x2=22, the second power, or square of 2. 2×2×2=23, the third power, or cube of 2. 2×2×2×2=24, the fourth power, or bi-quadrate of 2. 2×2×2×2×2=25, the fifth power of 2. 2×2×2×2×2×2=2o, the sixth power of 2. The exponent then shows how many times the root is to be a factor in the power required, and is always 1 greater than the number of multiplications producing the power. This use of the exponent differs from its use in algebra, where it is annexed to the power, and shows how many times the root is a factor in it. 4. The units, tens, &c., of any number may be involved separately; and their sum will equal the required power. The only use of this mode of involution is to show distinctly what a power is composed of. Illustrations. 1st. 14=10+4; and 14a=14x14=10+4 × 10+4=196. 10+4 4x10+4x4 10x10+4×10 10x10+2x4x10+4x4=102+2x4x10+42=100+80+16= 196. Thus, we obtain as the product of the units of the multiplier into the units of the multiplicand, 4x4-16; as the product of the units of the multiplier into the tens of the multiplicand, 4×10=40; as the product of the tens of the multiplier into the units of the multiplicand, 10x4=40; and as the product of the tens of the multiplier into the tens of the multiplicand, 10×10=100. We see then, that the second power, or square of 14, equals; 1st, the product of the first part into itself, 10×10; 2d, twice the product of the two parts into each other, 2×4×10; 3d, the product of the second part into itself, 4x4. 2d. So, 143=14×14×14=10+4×10+4×10+4=2744. 10+4×10+4=10×10+2×10×4+4×4, which being multiplied by 10+4 gives +2×4×4×10+-4×4×4 10×10×10+2×10×4×10+4×4×10 10×10×10+3×10×10×4+3×10×4×4+4×4×4= 103+3×102×4+3×10×42+43=1000+-1200+ 480+64 2744. It will be observed, that this product is composed of the cube of 10; three times the `product of the square of 10 into 4; three times the product of 10 into the square of 4; and the cube of 4; or, generally, the cube of the first part; three times the product of the square of the first part into the second; three times the product of the first into the |