SECTION XXIV. MISCELLANEOUS EXAMPLES. 1. From 6ac5 ab+c2 take 3ac- {3ab — (c— c2) Ans. 3ac2ab+2c2+6 c. 2. Reduce xa y2 — (— x y2 + x2 — *) x y − x2 y ―y (xy-x2)}) to its simplest form. Ans. 2 x2y+ x3. 3. Reduce (a − b + c)2 — (a (c — a − b) — {b (a + b + c) — c (a − b — c)}) to its simplest form. Ans. 2 (a2+b2 + c2). 4. Reduce (x + a) a + y- {(y + b) (x + b) — y (x + a − 1) − (x + y) (ba)} to its simplest form. Ans. a2 b2. 5. Reduce (a2b2) c — (a — b) {a (b + c) — b (a — c)} — x2. 11. Divide 118 x281 x by 1+6x+9x2. 12 Divide 9 a2+1 6x+9x 2 4 a1 6 a by 1+2 a2 13. Divide 9x — 7 x2 y2 + 2 y3 by 3x + 2x2 y — y2. 14. Divide 23 a 30-7a86 a1 by 3 a a 2 15. Find the prime factors of t2 — v [a2 + b2) (a + b) (2·7) M. 62m+ 16. Find the prime factors of 4 m n2 49 m2, n10. 17. Find the prime factors of x-2xy + y2. the 18. Find the prime factors of x3 7x444 prime factory of a Ex 4) (x-4) 19. Find the greatest common divisor of 5 - 10 x2 y +15 y3 and 4x3 + 8 x2 y + 8 x y2 + 4 y3. Ans. xy. 20. Find the greatest common divisor of 8 a b5 + 24 a b1 +16 a b and 766 + 7 65+ 7 b — 7 b2. Ans. b2b. 21. Find the greatest common divisor of 6 x2 + 7 x y − 3y2 and 12x2 + 22 x y + 6 y2. 2x + 3y 22. Find the greatest common divisor of 4 + 4x3 40 25. Reduce a2 b2 to its lowest terms. x+y х-у 26. Find the least common denominator and reduce 29. Reduce to one fraction with the least possible de 30. Reduce to one fraction with the least possible de 32. Reduce to one fraction with the least possible de 46. Reduce (a + x) (b − x) — a (b — c) — — 3, and c = 1. What is the value of > a 49. Find the value of x in the equation x = in its simplest form. a+b 50. A man spends $2. He then borrows as much money as he has left, and again spends $2. Then borrowing again as much money as he has left, he again spends $2, and then has nothing left. How much money did he have at first? 3.50 51. If 5 is subtracted from a certain number, two thirds of the remainder will be 40. What is the number? موت 52. Having a certain sum of money in my pocket, I lost c dollars, and then spent one ath part of what remained and had left one bth part of what I had at first. What was the original sum? What does the answer be 53. If I buy a certain number of pounds of beef at $0.25 a pound, I shall have $0.25 left; but if I buy the same number of pounds of lard at $0.15 a pound, I shall have $1.25 left. How much money have I? 2175 54. Divide 84 into three parts so that one third of the first, one fourth of the second, and one fifth of the third shall be equal. 21 28 35 55. In a certain orchard 25 more than one fourth of the trees are apple trees, 2 less than one fifth are pear trees, and the rest, one sixth of the whole, are peach trees. How many trees are there in the orchard? 60 56. A merchant spent each year for three years one third of the stock which he had at the beginning of the year; during the first year he gained $600, the second $500, and the third $400. At the end of the three years he had but two thirds of his original stock. What was his original stock? 2700 57. From a cask of wine out of which a third part had leaked, 84 liters were drawn, and then the cask was half full. What is the capacity of the cask? 504 The 58. A gentleman has two horses and a chaise. chaise is worth a dollars more than the first horse and b dollars more than the second. Three fifths of the value of the first horse subtracted from the value of the chaise is the same as seven thirds of the value of the second horse subtracted from twice the value of the chaise. What is the value of the chaise and of each horse? What are the 22 horse answers if a = 50 and b 50? chaise 200 150 12t horse 250 59. A had twice as much money as B. A gained $30 and B lost $40. Then A gave B three tenths as much as B had left, and had left himself 20 per cent more than he had at first. How much did each have at first? 608: 120 A's |