The greatest common measure of the two terms of this fraction was found in page 114 to be a-y; therefore, dividing both numerator and denominator by this quantity, we obtain as our result the fraction in its lowest terms; or, the greatest common measure of the two terms is found to be 2 a2+ 2 a b —¿3; and dividing both numerator and denominator by this quantity, the reduced fraction is, 4a3ex-4a3dx + 24a2bcx —24a"bdx+36ab'cx—3ab'dx (5.) Reduce Tabc7abdx3+7ac2r3_7acdx3- 216dx+216 ca+21bc2x2+21beda to its lowest terms. 4 a (a + 3b) Ans. 7x (b+c) 38. It frequently happens, however, that when the polynomials which form the numerator and denominator of a fraction which can be decomposed are not very complicated, we are enabled by a little practice to detect the factor and effect the reduction, without performing the operation of finding the greatest common measure, which is generally a tedious process. The results to which we called the attention of the reader, at the end of algebraic division (see page 107), will be found particularly useful in simplifications of this nature. Thus for example: (8.) Sa+8ab 5a3+10a2b+5ab5a(a2+2ab+b2)_ 5a(a+b)3 8a2(a+b) 8a2 (a+b) 73-73 — (a2 +ax+x2)(a—x) — a2+ax+z2 = = (a) ac+bd+ad+bc (10.) (a+b)c+(a+b)d = af+2bx+2ax+bf (a+b)ƒ+2x(a+b) F+2x (11.) = 6ac+10bc+9ad+15bd 3a(2c+3d)+5b(2c+3d) II. To reduce a mixed quantity to an improper fraction. 39. RULE.-Multiply the integral part by the denominator of the fraction, and to the product add the numerator with its proper sign; then the result placed over the denominator will give the improper fraction required. Thus, 40. It is to be remarked that when a fraction has the sign - it signifies that the whole fraction is to be subtracted, and consequently the negative sign applies to the numerator alone; and when the numerator is a polynomial, the negative aign extends to every term of the polynomial; thus, = 2 b c — ( b2 + c2—a2) 2bc a2 — (b2 — 2bc+c) 26c a1 — (b —c)2 III. To reduce fractions to others equivalent, and having a common denomi nator. 41. RULE-Multiply each of the numerators, separately, into all the denominators, except its own, for the new numerators, and all the denominators together for a common denominator. Thus: reduce and to equivalent fractions having a common denominator. ax d is the new numerator of the first, Therefore the fractions required are ad and bd b c b d Reduce 풍, to a common denominator. b d f h ln' adfhln cbfhin eb dhl n mbdfhl to a common denominator. are the fractions required. ADDITION OF FRACTIONS. 42, RULE.-Reduce the fractions to a common denominator, add the numerators together, and subscribe the common denominator. Thus: 43. RULE.-Reduce the fractions to a common denominator, subtract the numorator or the sum of the numerators of the fractions to be subtracted, from the numerator or the sum of the numerators of the others, and subscribe the common denominator. 44. When the denominators of the fractions which it is required to reduce are expressed in numbers, the result will frequently be much simplified by finding the least common multiple of the denominators, and then reducing the fractions to their least common denominator, according to the method explained in Arithmetic. Thus, if we are required to reduce the following fractions: The least common multiple of 4 and 5 is 20, the denominator of the third fraction; therefore the fractions, when reduced to their least common denominator, are the least common multiple of 3, 4, 6 is 12, which will be the least common denominator, and the above fractions become 45. RULE.—Multiply all the numerators together for a new numerator, and alt the denominators together for a new denominator. Thus, |