EXAMPLE. the ratio of receive? -$400 are to be divided between A, B, and C, in to A, į to B, and to C; how much will each of 400 200, and 200 X 400 ÷ 500 $160 A's share. B's share. ALLIGATION Medial is a method by which to find the mean price of a mixture or compound, consisting of two or more articles or ingredients, the quantity and price of each being given. RULE. Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities; the quotient will be the price per unity of measure of the mixture; and, having found the price of the given quantities as mixed, any quantities of the same materials, taken in like proportions, will be at the same price. EXAMPLE. If 20 lbs. of sugar at 8 cents, 40 lbs. at 7 cents, and 80 lbs. at 5 cents per pound, be mixed together, what will be the mean price, or price per pound, of the mixture? 20 X 8: = 160 40 X 7= = 280 140 400 ) 840 (6 cents. Ans. The several kinds, then, at their respective prices, taken in the proportion of 1 at 8, 2 at 7, and 4 at 5 cts., will form a mixture worth 6 cts. a pound. - EXAMPLE. If 10 lbs. of nickel are worth $2, and 24 lbs. of copper are worth $41, and 8 lbs. of zinc are worth 40 cts., and 1 lb. of lead is worth 5 cts., what are 5 lbs. of pretty good German silver worth? (200+ 450+40 +5) X 5 81 cents. Ans. 43 ALLIGATION Alternate is a method by which to find what quantity of each of two or more articles or ingredients, whose prices or qualities are given, must be taken to form a mixture or compound that shall be at a given price or of a given quality between the two extremes. It also applies to the finding of relative quantities when the quantity of one or more of the articles is limited. RULE. -Connect the given prices or qualities -a less than the given mean with that one or either one that is greater-and to the extent that all be thus connected; then place the difference between each given and the given mean opposite, not the given, or the given mean, but the given with which it is alligated; the num ber standing opposite each price or quality will be the quantity that must be taken at that price, or of that quality, to form a mixture or compound at the price or of the quality desired. And, being proportions respectively to each other, they may be taken in ratio greater or less, as desired. EXAMPLE. In what proportions shall I mix teas at 48 cents a pound and 54 cents a pound, that the mean price may be 50 cents a pound? In the proportions EXAMPLE. 2 lbs. at 54 cts. -In what proportions shall I mix teas at 48, 54, and 72 cents a pound, that the mixture may average 60 cents a pound? 48 60 541 72 12, 12, 12 + 6, EXAMPLE. A wine dealer has received an order for a quantity of wine at 50 cts. a gallon. He has none ready manufactured at that price. He has it at 40 cts., at 56 cts., and at 80 cents a gallon, and he has water that cost him nothing. He wishes to fill the order with a mixture composed of the four materials - the water and the three different priced wines. In what proportions must he mix them, that the mean or average price may be 50 cents? If, now, having found the proportions desired, it is wished to limit one of the articles in quantity-say the best wine to 8 gallons in the If, instead, it is desired to mix a given quantity, say 100 gallons, and proportioned, say as in first example, the quantity to be taken of each is ascertained by the following RULE. As the sum of the relative quantities is to the quantity required, so is each relative quantity to the quantity required of it respectively. The sum of the relative quantities alluded to is 6 +30 + 50 + 10 -96; then, 96 1006 = 61 INVOLUTION Consists in involving, that is, in multiplying a number one or more times into itself. The number so involved is called the root, and the product arising from such involution, its power. The second power, or square, of the root, is obtained by multiplying the root once into itself, as 4 X 4= 16; 4 being the root and 16 its square. The third power, or cube, of a number, is obtained by multiplying the number twice into itself, as 4 X 4 X 4-64; and so on for any power whatever. When a number is to be involved into itself, a small figure called the index or exponent is placed at its right, indicating the number of times it is to be so involved, or the power to which it is to be raised. Thus, 313 X 3 X 3 X 3=81; and 43 = 4 × 4 X 4 =64. EVOLUTION. EVOLUTION is the opposite of Involution. It consists in finding a root of a given number, instead of a power of a given root. When the root of a number is required or indicated, the number is written with the before it and the character or denomination of the root, if it be other than the square root, is defined by an index figure placed over the sign. When the square root of a number is required, the sign (✔) is placed before the number, but the index (2) is usually omitted. Thus, 25, shows that the square root of 25 is required, or to be taken; and 25 shows that the cube root is required. The operation is usually called extracting the root. TO EXTRACT THE SQUARE ROOT. RULE 1. Separate the given number into periods of two figures each, by placing a point over the first figure, third, fifth, &c., counting from right to left- the root will consist of as many figures as there are periods. 2. Find the greatest square in the left hand period, and place its root in the quotient; subtract the square of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Multiply the root so far found-the figure in the quotient - by 2, for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result (the number of times it is contained) in the quotient, to the right of the figure already there, and also to the right of the divisor; multiply the divisor, thus increased, by the last figure in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend. 4. Multiply the quotient- the root so far found (now consisting of two figures) — by 2, as before, and take the product for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result in the quotient, and to the right of the divisor, as before; multiply the divisor, as it now stands, by the figure last placed in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend, as before. 5. Multiply the quotient (now consisting of 3 figures) by 2, as before, and take the product for a divisor, and in all respects proceed as when seeking for the last two figures in the quotient. The quotient, when all the periods have been brought down and divided, will be the root sought. NOTE. 1. If there is a remainder after finding the integer of a root, annex periods of ciphers thereto, and proceed as when seeking for the integer. The quotient figures will be the decimal portion of the root. 2. If the given number is a decimal, or consists of a whole number and decimal, point off the decimal from left to right, by placing the point over the second, fourth, sixth, &c., figures therein, and fill the last period, if incomplete, by annexing a cipher. 3. If the dividend does not contain the divisor, a cipher must be placed in the quotient, and also at the right of the divisor, and the next period brought down; then the dividend must be divided by the divisor as increased. 4. If the quotient figure, obtained by dividing by the double of the root, is too large, as will sometimes be the case, (see 3d Example) it must be dropped, and a less-one which is the true measure- taken in its stead. EXAMPLE.-Required the square root of 123456.432. 123456.4320 (351.3636+. Ans. 9 Required the square root of 10621. Also, of 28561. RULE 1. Separate the given number into periods of three figures each, by placing a point over the first, fourth, seventh, &c., counting from right to left-the root will consist of as many figures as there are periods. 2. Find the greatest cube in the left hand period, and place its root in the quotient; subtract the cube of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Multiply the square of the quotient by 300, for a divisor; see how many times the divisor is contained in the dividend, and place the result (except that the remainder is large, diminished by one or two units) in the quotient. 4. Multiply the divisor by the figure last placed in the quotient, and to the product add the square of the same figure, multiplied by the other figure, or figures, in the quotient, and by 30; and add also thereto |