31. What two numbers are those whose product, difference of their squares, and quotient of their cubes are all equal? Ans. +5, and +5. 32. The product of two numbers is 10, and the product of their sum by the sum of their squares is 203; required the numbers found by a quadratic equation? Ans. 5 and 2. 33. There are three numbers in geometrical progression, the difference of whose differences is 6, and the sum of the numbers 42; what are the numbers? Ans. 24, 12, and 6. 34. There are three numbers in harmonical proportions, the difference of whose differences is 2, and the product of the extremes 72; what are the numbers? Ans. 6, 8, and 12. 35. Required three equidifferent numbers, such that if the first be increased by 1, the second by 2, and the third by the first, the sums may constitute an harmonical progression; but if 3 be added to the second, the sum may be a mean proportional between the sum of the numbers and the first diminished by ? Ans. 5, 6, and 7. 36. Required two such squares, that their difference shall be to the square root of the less as 3 to 7, and the square roots of the numbers to each other as 5 to 2? Ans. (19), and (4). 37. Given the sum of the cubes of two numbers =35, and the sum of their 9th powers =20195; what are the numbers? Ans. 3 and 2. 39. What two numbers are those, whose difference is 4, and their product multiplied by the sum of their squares 480? Ans. 6 and 2. 40. Given xxy=a, xz3√✓/xz=b, y°z/zy=c, to find the values of x, y, and z. 41. Required the values of x, y, and z, from the equations x(x+y+2)=a, y(x+y+2)=b, z(x+y+2)=c. 42. There are three numbers in harmonical proporportion, the sum of the first and third is 18, and their continued product 576, what are the numbers? Ans. 6, 8, and 12. 43. In how many different ways is it possible to pay £1000, without using any other coins than crowns, guineas, and moidores; a crown being 5s., a guinea 21s., and a moidore 27s.? Ans. 70734. 44. Given 2+y2+z2=266.5, x2+y+z2=176.5, (x+y+z)y=280. Required the values of x, y, and z. Ans. x=10.5, y=10, z=7.5 45. Required two cube numbers, such that the first multiplied into the product of their roots, shall be equal to the second; but the second multiplied into the product of their roots, shall be equal to 64 times the first? Ans. 8 and 64. 46. There are three numbers in geometrical progression, such that if the third be diminished by the first, the first, second, and the remainder, shall be in arithmetical progression; but the first and third being each increased by the second, and the second increased by 2, the numbers will be in harmonical progression; what are the numbers? Ans. 1, 2, 4. 47. Given x+y-x-y=249740, xy+x+y=8516, to determine the values of x and y. Result, x=500, y=16. 48. Given +y1+x2+y2=238632-2x2y', y2+z++z+y+2y°z=1640, x2+y3+z2=√(275100—x2—y3—zo,) to find the values of x, y, and z. Ans. x=22, y=2, z=6. 49. The sum of five numbers in geometrical progression is 242, and the fourth difference is 2; required the numbers? Ans. 2, 6, 18, 54, and 162. 50. Given x+y+z+v+w=12.15=a, b—x—y±√✓/{(b—x—y)3—4 (c—x—y)} 2 a—x—y—z±√✓{ (a—x—y—z)3—4(b—x—y—z)} w=a―x-y—z—v. 2 From the ambiguity of the signs, it is manifest that the number of numerical values of the unknown quantities go on increasing from a to v. THE END. the numbers will be in harmonical progression; what are the numbers? Ans. 1, 2, 4. 47. Given x+y-x-y=249740, xy+x+y=8516, to determine the values of x and y. Result, x=500, y=16. a―x—y—z±√ { (a—x—y—z)3—4(b—x—y—z)} w—a—x-y―z—v. 2 THE END. |