Simplify a + b + с (a-ba-c)(b−c)(b-a)* (c-a) (c—b) * The beginner should pay particular attention to this example. He is very liable to take the product of the denominators for the common denominator, and thus to render the operations extremely laborious. The second fraction contains the factor b-a in its denominator, and this factor differs from the factor a-b, which occurs in the denominator of the first fraction, only in the sign of each term; and by Art. 135, Also the denominator of the third fraction can be put in a form which is more convenient for our object; for by the Rule of Signs we have (c-a) (c-b)=(a–c) (b−c). Hence the proposed expression may be put in the form and in this form we see at once that the L. C. M. of the denominators is (a−b) (a−c) (b−c). By reducing the fractions to the lowest common denominator the proposed expression becomes that is a (b-c)-b(a-c)+c(a-b) (a-b)(a-c)(b-c) ab-ac-ab+bc + ac-bc (a - b) (a — c) (b −c) that is 0. 143. In this Chapter we have shewn how to combine two or more fractions into a single fraction; on the other hand we may, if we please, break up a single fraction into two or more fractions. For example, 51. 52. 53. 54. a-2b a-b с :—a) (a— (x − a) (a−b)' (x —b)(b· + (x − a) (a−b) 1 (x —b) (b− a)' a2 + b2 (x − a) (a − b) ̄ (x —b) (b− a) * 1 + 1 (a−b) (a–c) (b− a) (b−c) (a−b) (a−c) (b−a)(b· 144. Rule for the multiplication of fractions. Multiply together the numerators for a new numerator, and the denominators for a new denominator. 145. The following is the usual demonstration of the Rule. Let and be two fractions which are to be d multiplied together; put, andy; therefore And ac is the product of the numerators, and bd the product of the denominators; this demonstrates the Rule. Similarly the Rule may be demonstrated when more than two fractions are multiplied together. |