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. 5. 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? Ans. 4, 4, 4, 9. be 6. It is required to mix different sorts of grain, at 56, 62, and 75 cents per bushel, so that the mixture may worth 60 cents per bushel. How much of each kind must be taken? Ans. 17, 4, 4. Besides the variety of answers which may be obtained by connecting the simples differently, an infinite number of solutions may be found, by combining the different ratios, as we will illustrate by the aid of the following question: the ingredient of that rate; but if there be several, their sum will be the quantity required. Repeat this Rule 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 conditions of the question. It is obvious, that any other number of pounds which are to each other as the numbers Let ABCD be a parallelogram having AB for its base and DE its altitude. If from C we draw CF perpendicular to the base AB, meeting it, produced at the point F, the fig EFCD will be a rectangle equivalent to the pamilele gram, since the triangle A ED is obviously equal to the triangle BFC. The base EF of the rectangle isl to A B, the base of the parallelogram. The as of the rectangle is found (PROs. L) by multiplying the its altitude, and since the parallelogram is als rectangle, and since its base and akitude are mainly equal to the base and altitude of the retangle, it flowe that the area of the parallelogram may be found by tiplying its base by its altitude y many square rods in apering board 16 feet rcle being given to find ken as a unit, the cirarly. The exact value the diameter has never e has been extended to Is. (Geometry, B. V, circle is known, its cirllowing er Continued Fractions, we found some of the approximate values of this ratio to be 3, 22, 333, 355, &c. This last value of 355 is true to six places of decimals. It may be easily retained in the memory by observing that if the first three odd numbers, 1, 3, 5, be duplicated, they will stand 113355. Now the first three figures give the denominator, and the other three give the numerator of the ratio. EXAMPLE.--What is the circumference of the earth, on the supposition that it is 8000 miles in diameter ? Ans. 3.1416 x 8000-25132.8 miles, nearly. 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. 5. 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? Ans. 4, 4, 4, 9. 6. 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 ? Ans. 17, 4, 4. Besides the variety of answers which may be obtained by connecting the simples differently, an infinite number of solutions may be found, by combining the different ratios, as we will illustrate by the aid of the following question: one ingredient of that rate; but if there be several, their sum will be the quantity required. Repeat this Rule 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 conditions of the question. It is obvious, that any other number of pounds which are to each other as the numbers PROBLEM II-To find the area of a parallelogram. PROBLEM I-To find the area of a rectangle. Suppose ABCD to D be a rectangle whose length is 5 feet, and width 3 feet. If we divide this rect angle into portions of C B one square foot each, by means of lines drawn parallel to the sides of the rectangle, we shall obtain 15 such squares; that is, the rectangle will contain 15 square feet. In this example there are 3 strips of 5 square feet in each, or 5 strips of 3 square feet each. So that the number of square feet is found by multiplying the number of feet in length by the number of feet in width. Hence, to find the area of a rectangle we have this RULE. Multiply the length by the width, and the product will denote the number of squares of the same kind as the measure used in estimating the sides of the rectangle. If the sides of the rectangle are measured in feet, the product will be the 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. 5. 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? Ans. 4, 4, 4, 9. 6. It is required to mix different sorts of grain, at 56, 62, and 75 cents per bushel, so that the mixture may EXAMPLES. 1. How many square feet in a floor which is 16 feet wide and 23 feet long? And how many yards of carpeting, one yard wide, will cover the floor? 23×16=376=the number of square feet. be Since in one square yard there are square feet, we find 3769-413 the number of yards of carpeting required. 2. In a table 5 feet 3 inches long, and 3 feet 2 inches wide, how many square inches? And how many square feet? Ans. { 2394 sq. inches. 3. In a rectangular field which is 13 rods long, and 7 rods wide, how many square rods? of an acre? And what part is it 91 sq. rods. Ans. { 19 of an acre. 4. How many square inches in a square board 10 inches on a side? Ans. 110 sq. inches. 5. Which is the greater, a square board of 9 inches on a side, or a rectangular one 12 inches long and 7 wide? The rectangular piece Ans. contains 9 square inches more than the square one. |