2 and Then Let P and P' denote any two quantities, and let D denote their greatest common divisor. Let Q' denote the quotients of P and P' by D. shall we have, P = Qx D, and P' Since and Q' have no common factor, their least common multiple is QxQ'; consequently the least com mon multiple of P and P', is Dx Qx Q'; or, since P = DQ, we shall have the least common multiple equal to P x Q'. That is, the least common multiple of two quantities is equal to one of the quantities multiplied by the quotient of the other, by their greatest common divisor. Hence, to find the least common multiple of two quantities, we have the following RULE. Find their greatest common divisor; divide one of them by it, and multiply the other by the quotient. EXAMPLES. 1. Find the least common multiple of a3 x3 and Their greatest common divisor is a least common multiple is, The greatest common divisor of the two, is 2x hence, the multiple required is, 3. Find the least common multiple of a2 + 7x + 12, and x2+8x + 15. Ans. (x+3)(x + 4) (x + 5). When there are more than two quantities, find the least common multiple of the first and second, then that of the result and the third; and so on, to the last. Find the least common multiples of the following: 4. Of 8a2, 12a3, and 20a+1. Ans. 120a1. 5. Of x2+5x+6, x2 + 2x 8, and 2+7x+12. Ans. (x2+2x-8) (x2+5x+6.) 8. Of 18x+(xy), 25x3 (x − y)2, and 9. Of 231, and 2+x-2. 12x(x - y)3. Ans. 900x5(x-- y)2. 10. Of 6x2 — x — 1, and 2x2+3x 2. CHAPTER IV. FRACTIONS. 52. IF the unit 1 be divided into any number of equal parts, each part is called a FRACTIONAL UNIT are fractional units. Thus, 1 1 1 1 , י יד 53. A FRACTION is a fractional unit, or a collection 1 3 5 α of fractional units. Thus 2' 4' 7' b' are fractions. 54. 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 frac α tion the denominator b, shows that 1 is divided b' into b equal parts, and the numerator a, shows that a of these parts are taken. The fractional unit, in all cases, is equal to the reciprocal of the denominator. 55. A DECIMAL FRACTION is one whose denominator is some power of 10. In such fractions, the numerator is expressed by means of the decimal point, which sig nifies 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, 34 79 56. An ENTIRE QUANTITY, is one which contains no fractional part. Thus, 7, 11, a3x, 4x2 quantities. 3y, are entire 57. A MIXED QUANTITY, is a quantity containing both entire and fractional parts. Thus, 7, 83, a + mixed quantities. a bx are 58. Let denote any fraction, and q any quantity b whatever. From the preceding definitions, denotes a b 1 10 is taken a times; also, denotes that is that aq b Hence, multiplying the numerator of a fraction by any quantity, is equivalent to multiplying the fraction by that quantity. We see, also, that any quantity may be multiplied by a fraction, by multiplying it by the numerator, and then dividing the result by the denominator. 59. 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, pxq, as would be obtained by dividing it first by one of the factors, p, and then dividing that result by the other factor, q. That is, = (*) = pq Hence, multiplying the denominator of a fraction by any quantity, is equivalent to dividing the fraction by that quantity. 60. Since the operations of Multiplication and Division are the converse of each other, it follows, from the preceding principles, That, dividing the numerator of a fraction by any quantity, is equivalent to dividing the fraction by that quantity; And, dividing the denominator of a fraction by any quantity, is equivalent to multiplying the fraction by that quantity. 61. Since a quantity may be multiplied, and the result divided by the same quantity, without altering the value, it follows that, both terms of a fraction may be multiplied by any quantity, or both divided by any quantity, without changing the value of the fraction. TRANSFORMATION OF FRACTIONS. 62. The transformation of a quantity, is the operation of changing its form, without altering its value. FIRST TRANSFORMATION. To reduce an entire quantity to a fractional form having a given denominator. Let a be the quantity, and b the given denomi 63 nator. We have, evidently, a = ab ; hence, the |