252. RULE OF THREE IN VULGAR FRACTIONS. RULE.-Prepare the fractions by reduction, if necessary, and state the questionby the general rule (198); invert the first term, and then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the new numerator, written over the new denominator, will be the answer required. QUESTIONS FOR PRACTICE. 4. If my horse and chaise be worth $175, and the value of my horse be that of my chaise, what is the value of each ? 7: 1753: $105 horse. f: 175 :: : $70 chaise. 5. A lends B $48 for t of a year; how much must Blend A of a year to balance the favor? Ans. $86.40. 6. A person owning of a farm, sells of his share for £171; what is the whole farm worth? MISCELLANEOUS. For miscellaneous exercises, let the pupil review Section IV. Part I. and also the following articles: 51, 52, 55, 56, 57, 58, and 59. peach 1. In an orchard the trees bear apples, es, plums, 30 pears, 15 cherries, and 5 quinces; what is the whole number of trees ? 1+1+=+&+3= then 50 and 1 50 x 12600, Ans. Ans. £380. 2. One half, of a school, and 10 scholars, make up the school; how many scholars are there? Ans. 60. S. There is an army, to which if you add, and ‡ itself, and take away 5000, the sum total will be 100000, what is the number of the whole army? Ans. 50400 men. 4. Triple, the half, and the fourth of a certain number are equal to 104, what is that number? Ans. 27. 5. Two thirds and g of a person's money amounted to $760; how much had he? $600. 6. A man spent of his life in England, in Scotlaud, and the remaining 20 years, in the United States; to what age did he arrive? Ans. 48 years. 7. A pule is in the mud, in the water, and 12 feet out of the water; what is its length? Ans. 70 feet. 8. There is a fish whose head is 1 foot long, his tail as long as his head and half the length of his body, and his body as long as his head and tail both; what is the length of the fish? Ans. 8 feet. 9. What number is that whose 6th part exceeds its 8th part by 20? Ans. 480. 10. What sum of money is that whose 3d part, 4th part and 5th part are $94? Ans. $120. 11. If to my age there added be One half, 1,3d and three times 3, Six score and ten their sum will be; What is my age? pray show it me. Ans. 66 years. 12. Seven eighths of a certain number exceeds four fifths, by 6; what is that number? REVIEW. 6. When are fractions said to have a common denominator? 7. What is the common multiple of two or more numbers ?-the least common multiple ?-a prime number?-the aliquot parts of a num ber?-a perfect number? Explain. 8. What is denoted by a vulgar fraction?(129) How is an improper fraction changed to a whole or mixed number ?(216)-a whole or mixed number to an improper fraction? 9. How is a fraction multiplied by a whole number?(219)-divided by a whole number? 10. How would you multiply a whole number by a fraction ?(222) -a fraction by a fraction? 11. How would you divide a whole number by a fraction?(225) -a fraction by a fraction? 12. How may you enlarge the terms of a fraction ?(229) How diminish them? 13. How would you find the greatest common divisor of two numbers? How reduce a fraction to its lowest terms? 14. How would you find a common multiple of two numbers?(236) -the least common multiple? 15. How are fractions brought to a common denominator ?(239)--to the least common denominator? 16. How are fractions of a higher denomination changed to a lower denomination ?(243)-into integers of a lower?-a lower denomination to a higher?-into integers of a higher? 17. Is any preparation necessary in order to add fractions ?(243) —— why must they have the sanie denominator? How are they added ? iHow is subtraction of fractions performed? How the rule of three? SECTION VIII. POWERS AND ROOTS. V Envolution. ANALYSIS. 253. Let A represent a line 3 feet long; if this length be multiplied by itself, the product, (33) 9, feet is the area of the square, B, which measures 3 feet on every side. Hence, if a line, or a number, be multiplied by itself it is said to be squared, or because it is used twice as a factor, it is said to be raised to the second power; and the line which makes the sides of the square is called the first power; the root of the square, or its square root. Thus the square root of B-9, is A=3. A B 254. Again, if the square, B, be multiplied by its root, A, the product, (93)27 feet, is the volume, or content, of the cube, A CE, p. 51, [61.] which measures 3 feet on every side. Hence if a line or a number be multiplied twice into itself, it is said to be cubed, or because it is employed three times as a factor, (3x3x3=27) it is said to be raised to the third power, and the line or number which shows the dimensions of the cube, is called its cube root. Thus the cube root of A C E 27, is A-3. D 255. Again, if the cube, D, be multiplied by its root. A, the product, (27×3=) 81 feet, is the content of a parallelopipedon, A CE, whose length is 9. feet, and other dimensions, 3 feet each way, equal to 3 cubes, A C E, placed end to end. Hence if a given number be multiplied 3 times into itself, or employed four times as a factor, (3×3×3×3=81) it is raised to the fourth power, or biquadrate, of which the gives number is called the fourth root. 256. Again, if the biquadrate, D, be multiplied by its duct, (813) 243, is the content of a plank, equal to laid down in a square form, and called the sursolid, or which A is the fifth root. root, A, the pro9 cubes, A CE. fifth power, of 257. Again, if the sursolid, or fifth power, be multiplied by its root A, the product, (2433) 729. is the content of a cube equal to 27 cubes. A CE, and is called a squared cube, or sixth power, of which A is the sixth root. 258. From what precedes it appears that the form of a root, or first power is a line, the second power a square, the third power a cube, the fourth power a parallelopipedon, the fifth power a plank, or square solid. and the sixth power a cube, and proceeding to the higher powers, it will be seen that the forms of the 3d, 4th and 5th powers are continually repeated; that is, the 3d, 6th, 9th, &c. powers will be cubes, the 4th, 7th, 10th, &c. parallelopipedons, and the 5th, 8th, 11th, &c. planks. The raising of power of numbers is called er. INVOLUTION. 259. The number which denotes the power to which another is to be raised, is called the index, or exponent of the powTo denote the second power of 3 we should write 32, to denote the 3d power of 5 we should write 53, and others in like manner, and to raise the number to the power required, multiply it into itself continually as many times, less one, as are denoted by the index of the power, thus: 3 -3 =3 first power of 3, the root. -9 second power, or square of 3. =27 third power, or cube of 3. =3×3×3×3=81 fourth power, or biquadrate of 3. -3x3x3 34 QUESTIONS FOR PRACTICE. 1. What is the fifth pow- [ 2. What is the second er of 6 ? power of 45? 260. The powers of the nine digits, from the first to the sixth inclusive, are exhibited in the following 49 64 81 729 Squares, or 24 powers 1 4 9 16 25 36 Cubes, or 3d powers Sursolids, or 5th pow. 2. Evolution. ANALYSIS. 261. The method of ascertaining, or extracting the roots of numbers, or powers, is called Evolution. The root of a number, or power, is a number, which multiplied by itself continually, a certain number of times, will produce that power, and is named from the denomination of the pow er, as the square root, cube root, or 2d root, 3d root, &c. Thus 27 is the cube or 3d power of 3, and hence 3 is called the cube, or 3d, root of 27. 262. The square root of a quantity may be denoted by this character, called the radical sign, placed before it, and the other roots by the same sign, with the index of the root placed over it, or by fractional indices placed on the right hand. Thus 9, or denotes the square root of 9, 3 92 27, or 27, denotes the cube root of 27, and 16, or 164 denotes the 4th root of 16. The latter method of denoting roots is preferable, inasmuch as by it we are able to denote roots and powers at the same time. Thus 8 signifies that 8 is raised to the second power, and the cube root of that power extracted, or that the cube root of 8 is extracted, and this root raised to the 2d power: that is, the numerator of the index denotes the power, and the denominator the root of the number over which it stands. 263. Although every number must have a root, the roots of but very few numbers can be fully expressed by figures. We can however by the help of decimals approximate the roots of all sufficiently near for all practical purposes. Such roots as cannot be fully expressed by figures are denominated surds, or irrational numbers. 264. The least possible root, which is a whole number, is 1. The square of 1 is (1×1=) 1, which has one figure less than the number em |