Their greatest common divisor is a―x. common multiple is, Hence, their least 2. Find the least common multiple of 2x 1 and 4x2 1. The greatest common divisor of the two, is 2x-1: hence, the required multiple is, 3. Find the least common multiple of 2+7x+12, and x2 + 8x + 15. Ans. (x+3)(x+4) (x+5). If there are more than two quantities, we find the least common multiple of the first and second, then that of this result and the third, and so on, to the last. Find the least common multiples of each of the following groups of quantities: 4. Of 8a2, 12a3, and 20a1. Ans. 120a1. 5. Of x2+5x + 6, x2 + 2x - 8, and x2+ 7x + 12. Ans. (x2+2x-8) (x2+5x+6). 6. Of x 1, x2 1, and x2 + 4x 5. Ans. 40xy(x2- y2). 7. Of 10x(x + y), 8y(x − y), and 5(x2 — y2). 8. Of 18x4(x - y), 25x3(x — y)2, and 12x(x − y)3. Ans. 900x5(x- y)3. 9. Of 31, and x2+x-2. Ans. x2 + 2x3 — x — 2. CHAPTER IV. FRACTIONS. I. DEFINITIONS AND PRINCIPLES. Definitions. 53. If the unit 1 is divided into any number of equal parts, each part is called a fractional unit. 1 1 1 1 2' 4' 7' b' Thus, are fractional units. 54. A fraction is a fractional unit, or a collection 1 3 5 α of fractional units. Thus, a are fractions. 55. Every fraction is composed of two parts: the denominator, which shows into how many parts the unit 1 is divided; and the numerator, which shows how many of these parts are taken. Thus, in the fraction, the denominator shows that 1 is divided into b equal parts, and the numerator shows that a of these parts are taken. The fractional unit, in all cases, is equal to the reciprocal of the denominator. 56. A decimal fraction is one whose denominator is some power of 10. In such fractions, the denominator is expressed by means of a decimal point, which signifies that the denominator is equal to 1, followed by as many O's as there are decimal places. Thus, the expressions, .034, .0079, are decimal fractions, equivalent to the fractions, 8, TOOOO. 79 57. An entire quantity, is one which contains no fractional part. Thus, 7, 11, a3x, 4x-3y, are entire quantities. An entire quantity may be regarded as a fraction whose denomi 58. A mixed quantity, is a quantity containing bx both entire and fractional parts. Thus, 7, 84, a + are mixed quantities. Principles. α 59. Let denote any fraction, and let q be any b quantity whatever. From preceding definitions, 1 a b indicates that the fractional unit is taken a times; 1o. Multiplying the numerator of a fraction by any quantity is equivalent to multiplying the fraction by that quantity. We infer from what precedes that we may multiply any quantity by a fraction, by first multiplying that quantity by the numerator of the fraction, and then dividing the result by the denominator. It is a principle of division that the same result will be obtained if we divide the quantity a by the product of two factors pq, that would be obtained by dividing it by one of the factors, p, and then dividing that result by the other factor, q, that is, Hence, the following principle: 2°. Multiplying the denominator of a fraction by any quantity, is equivalent to dividing the fraction by that quantity. Since the operations of multiplication and division are the reverse of each other, we have, from what has been shown, the following principles: 3°. Dividing the numerator of a fraction by any quantity, is equivalent to dividing the fraction by that quantity. 4°. Dividing the denominator of a fraction by any quantity, is equivalent to multiplying the fraction by that quantity. Because a quantity may be multiplied by any quantity, and the result divided by the same quantity without changing its value, we have the following principle : 5°. Both terms of a fraction may be multiplied, or divided, by any quantity without changing the value of the fraction. |