2 Here, the least common multiple of 4cx2, 2x, and Sacs ; (Art. 147), is Saco x2 ; then, Sac x 2 X Sa'b=2ac + 3a2b=Ca'bc 4cxa Bacara new numeraXy=4aco Xy=4acy 2x Sac? x2 X 5x2=X 5x2=524 8ac2 6a3bc Hence and are the 8ac2x2' Sacox?? 8acx2 fractions required. tors; 4acʻy 5x 1 Ex. 6. Reduce and to a comat 3 2.0 mon denominator. 30x2 2a2x - 2x3 3a + 3x Ans. and Gax+6x2 Gar+6x? Gax +6x2 х 2x+3 5.0 +2 Ex. 7. Reduce and to a com 3ab mon denominator. 6abx +9ab 5x2 + 2x Ans. and 3abx 3ab. Ex. 8. Reduce 3' mon denominator. 4x2 +42 Ans. 12r+12 3x2+6x +3 122-412 a 2c2 За — х2 and + b' d' common denominator. adx 2bc x Ans. and bdx' bdx 3abd bdx 2025 40-15 Ex. 10. Reduce and 7+ 54' 3x 2 to a common denominator. 6x9 30axy-10~2 y + 50y Ans. and 30xy 60xy-15xy 30xy a 36 5& Ex, 11. Reduce and ata and 4a2-4x3' 4a2 - 4x2? 4a --4x2 § Ans. a -x2 1 1 Ex. 12. Reduce and a2 + 2ax + x2 5y to the least common denominator. -24 a3 +ara + ax + x3 as-ax4 ta' x a' - ax4 t-a4 x 75 5ay +5xy -a x4 tra4x -25 § V. ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS. To add fractional quantities together. RULE, 154. Reduce the fractions, if necessary, to a common denominator, by the rules in the last case, then add all the numerators together, and under their sum put the common denominator; bring the resulting fraction to its lowest terms, and it will be the sum required. C 20 5x Ex. 1. Add and together. 3' 7 ? 9 2x X7X9=126x 126x +135x + 21x_282x 5x X 3X9=13500 189 139 X7 X3= 21x 932 is the sum required. 189 3X7X9= 189 + a 2a 56 Ex. 2. Add and together. Õ 36 4a 12ao b+8ab-+-1563 a X 36 X 4a=12026 12ab2 2a X6 X 4a= 8a2b 20a2b-+1563 -(dividing by 6) 56 x 36 xb= 1563 12ab2 20a2 +1562 is the sum b X 36 X 4a=12ab2 12ab quired. Or, the least common multiple of the denominators may be found, and then proceed, as in (Art. 153), It is generally understood that mixed quantities are reduced to improper, fractions, before we perform any of the operations of Addition and Subtraction. But it is best to bring the fractional parts only to a common denominator, and to affix their sum or difference to the sum or difference of the integral part interposing the proper sign. 30* Ex. 3. It is required to find the sum of a b 2ar and bt с 3.22 2ax bc + 2ax Here, a and 6+ 6 6 Then, (ab — 3x2) xc=abc - 3cx 2 numerators. (bc+ 2ax) xbbc-2abx с - bXc=bc= denominator. + bc & bc 2abx - 3cx? 2abx - 3era zatot bc bc is the sum required. Or, bringing the fractional parts only to a common denominator, Thus, 302 Xc=3cx2 numerators, 2ax Xb=2abx And bxc=bc common denominator. 3cx2 2abx 2abx - 3cx Whence a +6+ =a+b+1 bc be the sum. bc Ex. 4. It is required to find the sum of 5:0+ -2 2x - 3 and 4x 3 5x Here, ( *—2) X 5x=5x2 -- 10x numerators. (x-3) X3 =6x - 9 = And 3 X 5x=15x common denominator. 5x2-10x 6x -9 Whence 5x -- +4x =9x + 150 15x 5x +-10%, 9-6.0 5x2-16x+9 + =9x+ to the sum 150 153 15x required. 6x -9 9-62 Here, is evidently (Art. 128); 153 15.30 but we might change the fractions into other equivalent forms before we begin to add or subtract ; thus, the fractional part of the proposed quantity 2x - 3 4x may be transformed by changing the 5x signs of the numerator, (Art. 128), and the quantity 3-260 itself can be written thus, 4x+ : It is well to 5x keep this transformation in mind, as it is often necessary to make use of it in performing several algebraical operations. 3a2 2a b Ex. 5. Add and together. 26' 5 7. 105a +-28ab +1002 Ans. 706 |