5. It is required to find the product of b+ and bx α Reduce the mixed quantities, if there are any, to a fractional form; then invert the terms of the divisor and multiply the fractions together as in the last case. The sign of the quotient will be changed by changing the sign either of the numerator or denominator, but will not be affected by changing the signs of both the terms. 78. We will add but two propositions more on the subject of fractions. If the same number be added to each of the terms of a proper fraction, the new fraction resulting from this addition will be greater than the first; but if it be added to the terms of an improper fraction, the resulting fraction will be less than the first. Let the fraction be expressed by α b' and suppose a <b. Let m represent the number to be added to each term: then In order to compare the two fractions, they must be reduced to the same denominator, which gives for Now, the denominators being the same, that the greatest which has the greater numerator. merators have a common part ab, and the part ond is greater than the part am of the first, since ab+bm ab + am ; fraction will be But the two nubm of the secb> a: hence that is, the second fraction is greater than the first. If the given fraction is improper, that is, if a >b, it is plain that the numerator of the second fraction will be less than that of the first, since bm would then be less than am. • If the same number be subtracted from each term of a proper fraction, the value of the fraction will be diminished; but if it be subtracted from the terms of an improper fraction, the value of the fraction will be increased. Let the fraction be expressed by to be subtracted by m. a and denote the number By reducing to the same denominator, we have, Now, if we suppose a <b, then am <bm; and if am<bm, then will that is, the new fraction will be less than the first. that is, the new fraction will be greater than the first. CHAPTER IV. OF EQUATIONS OF THE FIRST DEGREE. 79. AN Equation is the algebraic expression of two equal quanties with the sign of equality placed between them. Thus, x = a + b is an equation, in which x is equal to the sum of a and b. 80. By the definition, every equation is composed of two parts, separated from each other by the sign The part on the left of the sign, is called the first member, and the part on the right, is called the second member; and each member may be composed of one or more terms. 81. Every equation may be regarded as the enunciation, in algebraic language, of a particular problem. Thus, the equation is the algebraic enunciation of the following problem : To find a number which, being added to itself, shall give a sum equal to 30. Were it required to solve this problem, we should first express it in algebraic language, which would give the equation Hence we see that the solution of a problem by algebra, consists of two distinct parts: viz., the statement, and the solution of an equation. |