251. SUBTRACTION OF FRACTIONS. ANALYSIS. 1. What is the difference between of a dollar and of a dollar? difference. evidently expresses 2 tenths more than 3 tenths; then is the 2. What is the difference between of a yard and § of a yard? Here we cannot subtract from, for the same reason that we could not add them (49). We therefore reduce them to a common denominator, (224), and then the difference of the numerators (9—8—1), written over 24, the common denominator, gives for the difference of the fractions. RULE. Prepare the fractions as for addition (250), and then the difference of the numerators written over the common denominator will be the difference of the fractions required. 252. RULE OF THREE IN VULGAR FRACTIONS. RULE.-Prepare the fractions by reduction, if necessary, and state the question by 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. 1. If oz. cost £7, what is yd. wide, will line 131 yards of cloth that is 24 yds. will loz. cost? oz. £ oz. Then, wide ? For miscellaneous exercises, let the pupil review Section IV. Part I. and also the following articles: 51, 52, 55, 56, 57, 58, and 59. 1. In an orchard the trees bear apples, peaches, plums, 30 pears, 15 cherries, and 5 quinces; what is the whole number of trees? 1+1+=+&+2= ; then 50 and 13-50 X12 600, Ans. 2. One half, of a school, and 10 scholars, make up the school: how many scholars are there? Ans. 60. 3. There is an army, to which if you add 1, 3, and itself, and take away 5000, the sum total will be 10000; 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. 27H son's legacies £257 3s. 4d.: what was the widow's share? Ans. £635 10god. 15. A man died, leaving heir, and in his will ordered, his wife in expectation of an that if it were a son, of the estate should be his, and the remainder the mother's; but if a daughter, the mother should have, and the daughter; but it happened that she had both, a son and a daughter, in consequence of which the mother's share was $2000 less than it would have been if there had been only a daughter; what would have been the mother's portion, had there been only a son? Ans. $1750. REVIEW. 1. What are fractions? Of how many kinds are fractions? In what do they differ? 2. How is a vulgar fraction expressed? What is denoted by the denominator (22)? By the numerator? 3. What is a decimal fraction? How is it expressed? How is it read? How may it be put into the form of a vulgar fraction? 4. What is a proper fraction ? an improper fraction? What are the terms of a fraction? What is a compound fraction ?-a mixed num ber 5. What is meant by a common divisor of two numbers?-by the greatest common divisor? 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 number?-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 frac tion ? 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) 7-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 (249) ?why must they have the same denominator? How are they added? How is subtraction of fractions performed? How the rule of three? SECTION VIII. POWERS AND ROOTS. 1. Envolution. ANALYSIS. 253. Let A represent a line 3 feet long; if this length be multiplied by itself, the product (3×3), 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. 254. Again, if the square, B, be multiplied by its root, A, the product (9x3), 27 feet, is the volume, or content, of the cube, A C E, which measures 3 feet on every side. Hence, if a line or a number be multiplied twice into itself, it is said to F be cubed, or because it is employed 3 times as a factor (3x3x327), 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. 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 (3X3X3X3=81), it is raised to the fourth power, or biquadrate, of which the given number is called the fourth root. 256. Again, if the biquadrate, D, be multiplied by its root, A, the product, (81X3) 243, is the content of a plank, equal to 9 cubes, A CE, laid down in a square form, and called the sursolid, or fifth power, of which A is the fifth root. 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 INVOLUTION. 259. The number which denotes the power to which another is to be raised, is called the index, or exponent of the power. To 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 32=3X3 33=3x3x3 =3, first power of 3, the root. 9, second power, or square of 3. 34=3×3×3×3=81, fourth power, or biquadrate of 3. |