9. Required the greatest common measure of the fraction х4 -- Захз 8a-x2 + 18ax ~~ 8a4 Ans. 2? + 2ux 2a. 23 - axa 8a-x +6Q? 10. Required the greatest common measure of the fraction 545 +10a4b + 5a/2 Ans. a+b. 236 + 2a2b2 + 2ab3 + 74 11. Required the greatest common measure of the fraction 605 +15a46 - 4Q?2 10a2bc2 Ans. 3a - 20%. . 9a'b --- 27a^bc - 6abca + 18bc3? CASE II. To reduce fractions to their lowest or most simple terns. RULE.—Divide the terms of the fraction by any number, or quantity, that will divide each of them without leaving a remainder; or find their greatest common measure, as in the last rule, by which divide both the numerator and denomina tor, and it will give the fraction required. EXAMPLES. abc oc? 1. Reduce and to their lowest terms. 50262 ax + 2012 aabc 22 Here Ans. And Ans. 5a72 56 ax + x2 at 30 cx + 22 2. It is required to reduce to its lowest terms. dc + arx Here cx + 22 ac tax ctx aạc + aʻx (aa ac tarx or * Whence c + x is the greatest commion measure ; and c + ac) the fraction required, a-c + a-x an cx + x2 х to a single dimension only, I divide the same into the parts x2 + 2ax, and-bx— Zab; which, by inspection, appear to be equal to (+-+-2aXat , and (x + 2a) b. Therefore 2+2a is a divisor to both the parts, and likewise to the whole, expressed by. (-+-2a) X(2-6); so that one of these two factors, if the fraction given can he reduced to lower terms, must also measure the numerator; but the former will be found to su.cceed, the quotient coming out 2c2 -- ax+bx— ab, exactly; whence the fraction itself is reduced to axt-bu-ab which is not reducible farther by 2-6, since the di-O vision does not terminate without a remainder, as upon trial will be found. This rule is sometimes of great utility, because it spares great labour, and is very expeditious in reducing several fractions.-Ed. .22 3 C 72x 3. It is required to reduce to its lowest terms. 202 +2bic +62 2? + 2x + 5)23 - 12x (a 203 +2522 + 6 x Whence & to b is the greatest common measure; and a +b) 2003 - 62x 22 bac the fraction required. 22 + 2x + 32 And the same answer would have been found, if 203 — bax had been made the divisor instead of 202 + 25x + 62. 204 a4 4. It is required to reduce to its lowest terms. ac5 a-23 ox2 + a2 Ans. 103 Ans. Ans. % 6a2 + 7ax 3x 5. It is required to reduce to its lowest 6a + 11ax + 3x~ За — ° terms. 3a + 2 203 - 162 - 6 6. It is required to reduce to its lowest 3.2? 248 9 terms. 9x5 + 2003 + 4x2 X: +1 7. It is required to reduce to 1584 2x3 + 10x9 XC + 2 3x3 + 2a + 1 its lowest terms. Ans. 5x2 + 0 +2 a da cada a2c2 + A 8. It is required to reduce to its 4a'd - 4acd 4acd - 2aca +- 2c3 lowest terms. ada t- cd ac2 03 Ans. 4ad 2c2 2 CASE III. To reduce a mixed quantity to an improper fraction. RULE.-Multiply the integral part by the denominator of the fraction, and to the product add the numerator, when it is affirmative, or subtract it when negative; then the result, placed over the denominator, will give the improper fraction required. Here 33 ac 3 x 5 + 2 15 +2 17 Ans. 5 5 b Ans. a? X2 2. Reduce a + to improper fractions. And a and a C a tx? a 5. Let x be reduced to an improper fraction. 2α 2ax 22 Ans. 2a 2x 77 명 6. Let 5+ be reduced to an improper fraction. 3x 17x by Ans. 3х -a-1 7. Let 1 be reduced to an improper fraction. 2a +1 Ans. X 22 . * XXX = X. In adding the numerator a2-x2, the sign -affixed to the fraction denotes that the whole of that fraction is to be > subtracted, and consequently that the signs of each term of the numerator must be changed when it is combined with 22; hence the impro X2--02-22 2.22 per fraction is ED, CASE IV. To reduce an improper fraction to a whole or mixed quantity. RULE.—Divide the numerator by the denominator, for the integral part, and place the remainder, if any, over the denominator, for the fractional part; then the two, joined together, with the proper sign between them, will give the mixed quantity required. EXAMPLES. 27 ax + a2 1. Reduce and to mixed quantities. 27 5 al =+a (ax t-ao) = x = a + Ans. to a 20 Ans. a - 2c 2. It is required to reduce the fraction whole quantity. .xx2. ab - 2012 3. It is required to reduce the fraction to a mixed ab quantity. 2a Ans. 1 b a2 + x2 4. It is required to reduce the fraction to a mixed a 5. It is required to reduce the fraction to a whole Y quantity Ans. 22 + xy + yo 10x2 – 5x + 3 to a 6. It is required to reduce the fraction 5x 3 mixed quantity Ans, 2x - 1+ 50 CASE V. To reduce fractions to other equivalent ones, that shall have a common denominator. 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.* EXAMPLES. a and to fractions that shall have a common b с denominator. Here a Xcr ac the new numerators. ac bcs a ac Xc=bc the common denominator. 72 Whence and and the fractions required. b. bc 2x b 2. Reduce and to equivalent fractions having a com C mon denominator. 2cm ab Ans. and ac a a+b 3. Reduce and to equivalent fractions having a 7 common denominator. ac ab +33 Ans. and bc be 3x 26 4. Reduce and d, to equivalent fractions having a 2a' 30' common denominator. 9ca. 4ab 6acd Ans. and 6ac' bac 6ac 3 28 4x 5. Reduce to fractions having a com4' 3' 5' mon denominator. 45 40x 60a t-48x Ans. and 60' 60 60 atx 6. Reduce and to fractions having a common 2' ing? x denominator. 7a2 7ах бах 622 14a + 14* Ans. and 14a -- 14x140 14X 14a 14x and at а Зr * It may here be remarked, that if the numerator and denominator of a fraction be either both multiplied, or both divided, by the same number or quantity, its value will not be altered; thus 2 2 X 3 6 3 3 : 3 1 ab and ; or and bc which method is often of great use in reducing fractions more readily to a common denominator. a ac 9 bci |