122 SINGLE RULE OF THREE IN VULGAR FRACTIONS. EXAMPLES. 1. If % of a yard of cloth cost $28, what will 56 yards cost at the same rate ? Ans. $16.25. Operation, 5=Hyds. yds. yds. $ $2]=udols. If }: if : * 8 x 65 x 21=10920 65 7x 12 x 8 = 672 4 )65. $16.25 Ans. 2. If a man drink daily a dram which costs 64cts, how much will he expend in this manner in 20 years of 3651 days each? Ans. $456.56,25. 3. What is the value of ß of $ of f of a pound, at the rate of of a dollar for of a pound? Ans. $17, or 42 cts. 4. How many yards of baize, ell English wide, will be sufficient to line 20 yards of camlet & yard wide? Ans. 12 yards. 5. What quantity of shalloon, 4 yard wide, will line 7} yards of cloth, 1 yd wide? Ans. 15 yards. 6. Bought 54 pieces of muslin, each pirce containing 274 yards, at of a dollar per yard-what did they cost! Ans. $ 5.50 7. If A can mow an acre of grass in 5 hours, and B can mow 1$ acres in 9} hours, in what time can they jointly cut 84 acres ? Ans. 224 hours, or 22 hours 40min. 8. If a coat and waistcoat can be made of 38 yards of broadcloth 18yd. wide, how many yards of stuff & yd. in breadth, will it require to fit the same person? Ans. Syds. 9. How much in length that is 7 in. broad, will make a foot square? Ans. 1817 inches. 10. How many pounds of potash can I purchase for 124 cents, if a pound cost 19 cents ? Ans. 7 pounds. 11. A gentleman owning of a vessel, sells of his share for $312-What is the whole vessel worth? Ans. $1300. 12. If of a gallon cost f of a dollar, what will of a ton cost ? Ans. #140. 13. If 3 men can nish a piece of work in 41 hours, how many hours will it require one man to do the same? Ans. 131 hours. INVOLUTION, OR THE RAISING OF POWERS. A POWER is the product arising from multiplying any given number into itself continually a certain number of times, thus: 3x 3=9=32, the 2d power or square of 3. 3x 3x 3=27=33, the 3d power or cube of 3. 3x3x3x3=81, the 4th power=34 or the biquadrate of 3, &c. The number denoting the power is called the index of the power. RULE. Multiply the number, root, or first power, continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required. Note.-Vulgar fractions may be involved to any power by raising each of the terms (numerator and denominator) to the power required; and if a mixed number be proposed, reduce it to an improper fraction, and then proceed as above directed. Or, A vulgar fraction may be reduced to a decimal, and then involved as a whole number. EXAMPLES. 1. What is the fifth power of 9? 9 81=2d power 729=3d do. 9 6561=4th do. 9 59049=5th do. or ans.=95 2. What is the square of 549 ? Ans. 301401. 3. Required the cube of 54.36 Ans. 160634.321856 4. Required the 8th power of .15 Ans. .0000002562890625 5. What is the 5th power of j? Ans. s. 6. What is the 5th power of 11? Ans. 1642 7. What is the 20th power of 2?. Ans. 1048576. NOTE.-In extensive involutions, the operations may be abridged by multiplying together two of the powers which have been obtained, instead of repeating the multiplication with the root. EVOLUTION, OR EXTRACTING OF ROOTS. The extracting of roots is the finding of numbers which raised to given powers will be equal to given numbers. Any power of a given number can always be found; but the roots of given numbers cannot be always accurately obtained ; thus we cannot find a number which being multiplied by itself will produce the number 5. But by the help of deci. mals we can approximate towards the root, to any requisits L2 126 SQUARE ROOT. degree of accuracy. The roots to such numbers, which are perfectly found, are called rational roots. The roots which approximate, that is, those that cannot be accurately expressed in numbers, are called surd roots : thus the square root of 5 is a surd root. The sign v prefixed to a number, denotes its root; thus, ✓ 5 denotes the square root of 5. V8 indicates the cube root of 8. 16 denotes the 4th root of 16, &c. If the root be expressed by several numbers with the sign + or between them, a line is drawn over all the parts of it: thus, the square root of 25+11=6; and the cube root of 820–91=9. EXTRACTION OF THE SQUARE ROOT. RULE. 1. Separate the given number into periods of two figures each, beginning at the unit figure, or decimal point. 2. Find the greatest square in the first (left nand) period, and set its root on the right hand of the given number, after the manner of the quotient figure in Division. 3. Subtract the square thus found, from the said period, and to the remainder, annex the second period, for a dividend. 4. Double the root already found, for a divisor, and find how often it is contained in the dividend, exclusive of the place of units; annex the result* to the quotient and also to the divisor. 5. Then multiply and subtract as in Division, and to the remainder, annex the third period for a new dividend. 6. To the (last) divisor, add the last figure of the root for a new divisor, find another quotient figure as before, annex it to the root, and the divisor, multiply and subtract as before. Note 1.-If there be decimals in the given number, it must be pointed both ways from the unit or decimal point, and the root will consist of as many figures of whole numbers and decimals respectively as there are periods of integers or decimals in the power. Note 2.—The root of a vulgar fraction is found by reducing it * The quotient found by this mode is sometimes too great: when this is the case, it must be diminished until the complete divisor, multiplied by the quotient figure, produces a result equal to or less than the dividend. |