THE DOUBLE RULE OF THREE, IN DECIMALS. Questions in this rule are wrought as in whole numbere, placing the points agreeably to former directions. EXAMPLES. 1. If 3 men receive 8.9L. for 19.5 days labour, how much must 20 men have for 100.25 days? men 3 : 20 } days 19.5: 100.25 days Ans. 305L. Os. 8.2d. :: 89L. : 305L. Os. 8.2d. 2. If 2 persons receive 4.625s. for 1 day's labour, how much should 4 persons have for 10.5 days? Ans. 4L. 17s. 1 d. 3. If the interest of 76.5L. for 9.5 months, be 15.24L. what sum will gain 6L. in 12.75 months? Ans. 22L. 8s. 93d. 4. How many men will reap 417.6 acres in 12 days, if 5 men reap 52.2 acres in 6 days? Ans. 20 men. 5. If a cellar 22.5 feet long, 17.3 feet wide, and 10.25 feet deep, be dug in 2.5 days, by 6 men, working 12.3 hours a day; how many days of 8.2 hours, should 9 men take to dig another, measuring 45 feet long, 34.6 wide, and 12.3 deep? Ans. 12 days. 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, 2x2= 4 the second power or square of 2. 2x2x2= 8 the third power or cube of 2. 2×2×2×2=16 the fourth power of 2, &c. The number denoting the power is called the index or exponent of that power. If two or more powers of the same number are multiplied together, their product is that power whose index is the sum of the exponents of the factors; thus, 2x2-4 the square of 2; 4×4=16=4th power of 2; and 16×16=256=8th power of 2, &c. 1953125 525 125 625 3125 15625 78125 390625 864 512 4096 32768 262144 2097152 16777216 134217728 981729 6561 59049531441 4782969 43046721 387420489 EXAMPLES. 1. What is the square of 22? Ans. 484. 2. What is the cube or third power of 4? Ans. 64. 4x4x4-64. 3. What is the fifth power of 7? 4. What is the cube or third power of 35? Ans. 16807. Ans. 42875. 5. What is the fourth power of ? Ans. 81 256' 6. What is the cube or third power of .13? Ans..002197 7. What is the sixth power of 5.03? Ans. 16196.005304479729 EVOLUTION, OR THE EXTRACTING OF ROOTS. The root of a number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus 2 is the square root of 4, because 2×2=4; and 4 is the cube root of 64, because 4×4×4=64, and so on. THE SQUARE ROOT. The square of a number is the product arising from that number multiplied into itself. Extraction of the square root is the finding of such a number as being multiplied by itself will produce the number proposed. RULE. 1. Separate the given number into periods of two figures, each, beginning at the units place. 2. Find the greatest square contained in the left hand period, and set its root on the right of the given number: subtract said square from the left hand period, and to the remainder bring down the next period for a dividual. 3. Double the root for a divisor, and try how often this divisor (with the figure used in the trial thereto annexed) is contained in the dividual: set the number of times in the root; then, multiply and subtract as in division, and bring down the next period to the remainder for a new dividual. 4. Double the ascertained root for a new divisor, and proceed as before, till all the periods are brought down. Note. If, when all the periods are brought down, there be a remainder, annex ciphers to the given number, for decimals, and proceed till the root is obtained with a sufficient degree of exactness. Observe that the decimal periods are to be pointed off from the decimal point toward the right hand; and that there must be as many whole number figures in the root, as there are periods of whole numbers, and as many decimal figures as there are periods of decimals. PROOF. Square the root, adding in the remainder, (if any,) and the result will equal the given number. 3. What is the square root of 451584? Ans. 672. 4. What is the square root of 36372961? Ans. 6031. 5. What is the square root of 7596796? Ans. 2756.228+ 6. What is the square root of 3271.4007? Ans. 57.19+ 7. What is the square root of 4.372594? Ans. 2.091+ 8. What is the square root of 10.4976? Ans. 3.24 9. What is the square root of .00032754? Ans..01809+ 10. What is the square root of 10? Ans. 3.1622+ To extract the Square Root of a Vulgar Fraction. RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator. Note. If the fraction be a surd, that is, one whose root can never be exactly found, reduce it to a decimal, and extract the root therefrom. To extract the Square Root of a Mixed Number. RULE. Reduce the mixed number to an improper fraction, and proceed as in the foregoing examples: Or, Reduce the fractional part to a decimal, annex it to the whole number, and extract the square root there from. EXAMPLES. 1. What is the square root of 378? 2. What is the square root of 27%? 3. What is the square root of 8514? 4. What is the square root of 8? APPLICATION. 1. The square of a certain number is 105625: what is that number. Ans. 325. 2. A certain square pavement contains 20736 square stones, all of the same size, what number is contained in one of its sides? Ans. 144. 3. If 484 trees be planted at an equal distance from each other, so as to form a square orchard, how many will be in a row each way? Ans. 22. 4. A certain number of men gave 30s. 1d. for a charitable purpose; each man gave as many pence as there were men: how many men were there? Ans. 19. Note. The square of the longest side of a right angled triangle is equal to the sum of the squares of the other two sides; and consequently the difference of the square of the longest, and either of the other, is the square of the remaining one. 5. The wall of a certain fortress is 17 feet high, which is surrounded by a ditch 20 feet in breadth; how long must a ladder be to reach from the outside of the ditch to the top of the wall? Ans. 26.24+feet. |