days. What is the distance between the two places? and how far had each man traveled? Ans. Distance, 100 miles; A had gone 60, B 40 miles. 44. The product of the two dimensions of a rectangular piece of land, subtracted from the square of the greater dimension, leaves 300 square rods, and subtracted from the square of the less, leaves 200 square rods. What are the dimensions of the piece? Ans. This problem is impossible; how is its impossibility indicated? and in what does this impossibility consist? 45. A person being asked the ages of his two children, replied, that the difference of their ages was 3 years, and the product multiplied by the sum of their ages was 308. What were their ages? This problem will result in an Affected Cubic Equation. Ans. 7 and 4 years. 46. A gentleman who had a square lot of ground, reserved 10 square rods out of it, and sold the remainder for $432, which was as many dollars per square rod as there were rods in a side of the whole square? What was the length of its sides? Ans. 8 rods. 47. A and B set out together from the same place, and travel in the same direction. A goes the first day 28 miles, the second 26, and so on, in arithmetical progression; while B goes uniformly 20 miles per day. In how many days will the two be together again? Ans. 9 days. 48. A farmer wishes to build a crib whose capacity shall be 1620 cubic feet, and whose length, breadth, and height shall be in an arithmetical progression decreasing by the common difference 3. What must be the dimensions of the crib? It may be well to remind the student here, that cubic measure, or measure of capacity, is found by multiplying together length, breadth, and height or depth. Ans. 15, 12, and 9 feet. 49. One traveler sets out to go from A to B, at the same time at which another sets out from B to A. They both travel uniformly, and at such rates, that the former, 4 hours after their meeting, arrives at B, and the latter at A, in 9 hours after. In how many hours did each one perform the journey? Ans. 10, and 15 hours. 50. A lady on being asked the ages of her three little boys, answered that they were in harmonical progression; and the sum of their ages was 22 years; and that if the ages of the two elder were each increased by of itself, the three would then be in geometrical progresWhat were the respective ages? Ans. 4, 6, and 12 years. sion. 51. A person wishes to construct two cubical reservoirs which shall differ in their linear dimensions by 4 feet, and which shall together contain 5824 cubic feet. What must be the dimensions of the two reservoirs? Ans. 12, and 16 feet 52. Two partners, A and B, divided their gain, which was $60, when B's share was found to be $20. A's capital was in trade 4 months; and if the number 50 be divided by A's capital, the quotient will be the number of months that B's capital, which was $100, continued in trade. What was A's capital? and the time B's was in Ans. A's Capital $50; B's 1 month in trade. 53. Let there be a square whose side is 110 inches; it is required to assign the length and breadth of a rectangle whose perimeter shall be greater than that of the square by 4 inches, but whose area shall be less than the area of the square by 4 square inches. trade? Ans. 126, and 96 inches. 54. There is a number consisting of three digits which increase from left to right by the common difference 2; and the product of the three digits is 105. Required the number.. Ans. 357. 55. A person bought 2 pieces of cloth for $63. For the first piece he paid as many dollars per yard as there were yards in both pieces, and for the second as many dollars per yard as there were yards in the first more than in the second; also the first piece cost six times as much as the second. What was the number of yards in each piece? Ans. 6, and 3 yards. 56. There is a number consisting of 4 digits which decrease from left to right by the common difference 2; and the product of the four digits is 945. Required the number. Ans. 9753. 57. A gentleman purchased two square lots of ground for $300; each of them cost as many cents per square rod as there were rods in a side of the other, and the greater contained 500 square rods more than the less. What was the cost of each lot? Ans. $180, and $120. 58. A merchant bought a number of bales of cloth. The number of pieces in each bale was 10 more than the number of bales, and the number of yards in each piece was 5 more than the number of pieces in each bale; and the whole quantity was 1500 yards. What was the number of bales? Ans. 5 bales. 59. A person dies, leaving children, and a fortune of $46800, which, by his will, is to be divided equally amongst them. Immedi ately after the death of the father, two of the children also die, in consequence of which each surviving one receives $1950 more than he was entitled to by the will. How many children did the father leave? Ans. 8 children. 60. A coach set out from Cambridge for London with 4 more outside than inside passengers. Seven outside passengers went at 2 shillings less than 4 inside ones, and the fare of the whole amounted to £9. At the end of half the journey, 3 more outside and one more inside passenger were taken up, in consequence of which the fare of the whole was increased in the proportion of 19 to 15. Required the number of passengers at first, and the fare of each. Ans. 5 inside, and 9 outside passengers; fares 18 and 10 shillings. 61. There is a fraction which, inverted and increased by, will be less than 2; and if its numerator be increased by 2, the value of the fraction will be greater than . What is the fraction? Ans. Between 3 and . 62. In a purse which contains 24 coins of silver and copper, each silver coin is worth as many pence as there are copper coins, each copper coin is worth as many pence as there are silver coins, and the whole is worth 18 shillings. How many were there of each kind of coins? Ans. 6, and 18. 63. The square root of a certain number plus 4, is less than 9; and ten times the square root of the number minus 2, is greater than eight times the square root of the number plus 4. What is the number? Ans. Between 9 and 25. 64. A and B traveled on the same road, and at the same rate, from Huntingdon to London. At the 50th mile stone from London, A overtook a drove of geese, which were procecding at the rate of 3 miles in 2 hours; and 2 hours afterwards met a wagon which was moving at the rate of 9 miles in 4 hours. B overtook the same drove of geese at the 45th mile stone, and met the same wagon 40 minutes before he came to the 31st mile stone. London? Where was B when A reached 221 CHAPTER XII. GENERAL THEORY OF EQUATIONS. STURM'S THEOREM-HORNER'S METHOD OF SOLVING THE HIGHER EQUATIONS. (280.) A GENERAL THEORY OF EQUATIONS consists of an exposition of the general properties of Equations. Properties relating to the Divisors of an equation (253), a criterion for Roots of an equation (254), the Number of roots (255), and the general Law of the Coefficients of an equation (267), have been presented in a former Chapter; and it is only necessary therefore to add here the more important of the remaining parts of this extensive theory. General Equation. (281.) Every complete Equation containing only integral powers of the unknown quantity x, may be reduced to the general form by transposing all the terms to one side, and dividing by the coefficient of the highest power of x. The exponent n is equal to the degree of the particular Equation, and the exponents decrease regularly by 1 in the successive terms. If any of the lower powers of x be wanting in the given Equation, those powers may be introduced by giving to each of them the coefficient 0-the Equation will thus be rendered complete. Thus the Equation 23+4x-10=0, when made complete is, x3+0x2+4x-10=0, or, simply, x3 +0 +4x-10=0. Some of the terms must necessarily be positive, and others negative, in the first member, to make the second member 0; but in the general Equation the sign + is prefixed to each term, to denote the algebraic sum of all the terms. Change of Signs of the Roots of Equations. (282.) If the signs of the alternate terms of any complete Equation be changed, the signs of all the roots will be changed. If we take the general Equation, (1) x2+axn−1 + bxn−2+ &c.=0; and change the signs of the alternate terms, we shall have When the exponent n is an even number, the terms containing the odd powers of x will be negative in Equation (2); and when n is an odd number, the terms containing the odd powers of x will be negative in Equation (3). If now the substitution of r for x in Equation (1) result in 0, the same result will be obtained by substituting -r for x in Equation (2) or (3); since the odd negative powers of -r, (202), in the latter Equations will become positive when taken subtractively, and these Equations will then be the same as Equation (1). Hence, if r be a root of Equation (1); - will be a root of Equation (2) or (3); that is, if the signs of the alternate terms, &c. The roots of the Equation 23-7x2+36 (x-6) (x-3)(x+2)=0, = are 6, 3, and −2, (254); what is the Equation whose roots are -6, -3, and 2? Completing the given Equation by introducing the first power of x, and changing the signs of the alternate terms, we have x3+7x2+0x-36 or x3+7x2-36=(x+6) (x+3) (x-2)=0. of which the roots are -6, -3, and 2, (254). From the principle thus demonstrated it follows, that (283.) The positive roots of any complete Equation become negative, and the negative ones become positive, by changing the signs of the alternate terms of the Equation. |