The results are the same; but observe that the fraction must not be reduced to the least common denominator, but only to their common denominator, the one multiplied by the other; of course, the mixtures would be of the same value, and each pair would have the same relative value, but the four would not correspond. REM.-Observe that different results may be obtained by different methods, and all will be correct. Sometimes a merchant wishes to put the whole of a certain kind into the mixture; in which case this article may be mixed with every kind that is on the opposite side of the mean. 4. A merchant has five qualities of liquors, at the following prices per gallon, viz., $1.25, $1.45, $1.60, $1.80, and $1.90, and he has an order for liquor at $1.50 per gallon. Of the liquor at $1.90 he has 40 gallons, all of which he wishes to put into the. mixture; how much must be taken of each of the other kinds? X 5. A grocer has spices at 18, 24, 36, and 42 cts. per lb. of which he wishes to make a mixture worth 32 cts. COR. 1.-Connect any two rates, one of which is less and the other greater than the mean. Take the difference of each of these and the mean, and place the difference opposite the price of the one with which it is. connected. COR. 2.-When there are odd terms, that is, when those above and below the mean price are not of equal numbers, the odd term will be connected with an opposite one that is already connected, and instead of having only one difference opposite to it, will have two or more; these several numbers must be added together, and their sum will be the quantity to be taken of the price to which it stands opposite. 6. How many pounds of each kind of tea, of the values of 30 cts., 35 cts., 40 cts., 52 cts., 55 cts., and 60 cts., must be taken to make a mixture worth 45 cts.? REM. The results will be correct, although the portions will be different by making other connections. INVOLUTION AND EVOLUTION Involution and Evolution correspond very nearly to Multiplication and Division. Involution consists in the multiplication of the same number by itself, or of the same factor entering two or more times into a product; whilst Evolution takes the product and restores the equal factors of which it is composed. Involution is called the raising of powers; thus, 2 × 2 = 4 is called the 2d power of 2. 2 × 2 × 2 = 8 is called the 3d power of 2. 3 x 39 is called the 2d power of 3. 3 × 3 × 3 = 27 is called the 3d power of 3. Evolution, or the extracting of roots, is exactly the reverse of Involution. In order to extract the second or square root, we take the second power and from it restore the factors; thus, 4 is the second power of 2; hence the square root of 4 is 2; and the square root of 9 is 3; of 16 it is 4; of 25 it is 5; of 36, 6; of 49, 7; of 64, 8, etc. The number given is a square surface, each side of which is a factor; the sides of the square being equal, the factors are equal; thus, D The figure ABCD is a square surface, 3 inches in length and 3 inches in breadth; the angles at A, B, C, and D are equal, that is, from each of these points the sides have the same divergence; hence, if any one be placed on the other, the two will coincide. B AB is 3 inches long, and for every inch in breadth there are 3 square inches, and since AD and BC are each 3 inches, there will be 3 times 3 sq. in. = 9 sq. in. In the square root we have given the area of the square surface to find a side; and as 3 × 3 = X 9, we know that 9 is composed of the two equal factors of 3. .. 3 is the square root of 9. There are many numbers of which we cannot obtain an exact root, as we can only extract the root exactly of 4, 9, 16, 25, 36, 49, 64, 81, 100, in 100; that is, 9 have exact roots and 91 have not. Of these we can approximate decimally, or we can put them under the radical sign, thus, √3, √5, √6, by which the root is expressed, and they are read, the square root of 3, or 5, or of 6, as the case may be. In larger numbers, it is more difficult to get the root; thus, 11 x 11 = 121; 12 × 12 = 144; 13 × 13 = 169, etc. For the extraction of these roots a method will be given hereafter. Sometimes, however, the number is not a perfect square, and hence only an approximate root can be found. The square of the highest digit has two places of figures, the square of the least one; the square of the least number of tens is hundreds, and when the tens reach 4 it has four places of figures, never more. An increase of one figure in the root makes an increase of two in the square, for the square of the least unit is a unit, and of the least units and tens is hundreds; the square of the largest digit is tens, and the square of the largest tens is thousands. Therefore, if the square be pointed off in periods of two figures each, there will be as many periods as figures in the root. |