49. A man rows down a stream, whose rate is 35 miles per hour, for a certain distance in 1 hour and 40 minutes. In returning, it takes him 6 hours and 30 minutes to arrive at a point 2 miles short of his starting-place. Find the distance which he rowed down stream, and his rate of pulling. 50. If a certain number be divided by the sum of its two digits, the quotient is 6 and the remainder 1. If the digits be inverted, the quotient of the resulting number increased by 8, divided by the sum of the digits, is 6. Required the number. 51. A train running from A to B meets with an accident which delays it 30 minutes; after which it proceeds at threefifths its former rate and arrives at B 2 hours and 30 minutes late. If the accident had occurred 30 miles nearer A, the train would have been 3 hours late. What was the rate of the train before the accident? 52. A, B, and C together have $24. A gives to B and Cas much as each of them has; B gives to A and Cas much as each of them then has; and C gives to A and B as much as each of them then has. They have now equal amounts. How much did each have at first? 53. A and B are building a fence 126 feet long. After 3 hours, A leaves off, and B finishes the work in 14 hours. If 7 hours had occurred before A left off, B would have finished the work in 4 hours. How many feet does each build in one hour? 54. Divide 115 into three parts such that the first part increased by 30, twice the second part, increased by 2, and 6 times the third part, increased by 4, may all be equal to each other. 55. Four men, A, B, C, and D, play at cards, B having $1 more than C. 56. A gives to B and C as much as each of them has; B gives to A and C as much as each of them then has; and C gives to A and B as much as each of them then has. Each has now $48. How much did each have at first? 57. A, B, and C, were engaged to mow a field. The first day, A worked 2 hours, B 3 hours, and C 5 hours, and together they mowed 1 aère; the second day, A worked 4 hours, B 9 hours, and C 6 hours, and all together mowed 2 acres; the third day, A worked 10 hours, B 12 hours, and C 5 hours, and all together mowed 3 acres. In what time could each alone mow an acre? 58. A man invests $3600, partly in 34 per cent bonds, and partly in 4 per cent bonds. The income from the 3 per cent bonds exceeds the income from the 4 per cent bonds by $6. How much has he in each kind of bond? 59. A and Brun a race of 480 feet. The first heat, A gives B a start of 48 feet, and beats him by 6 seconds; the second heat, A gives B a start of 144 feet, and is beaten by 2 seconds. How many feet can each run in a second? 60. The fore-wheel of a carriage makes 4 revolutions more than the hind-wheel in going 96 feet; but if the circumference of the fore-wheel were as great, and of the hind-wheel as great, the fore-wheel would make only 2 revolutions more than the hind-wheel in going the same distance. Find the circumference of each wheel. 61. A and B together can do a piece of work in 4 days; but if A had worked one-half as fast, and B twice as fast, they would have finished it in 4 days. In how many days could each alone perform the work? 62. A and B run a race of 300 yards. The first heat, A gives B a start of 40 yards, and beats him by 2 seconds; the second heat, A gives B a start of 16 seconds, and is beaten by 36 yards. How many yards can each run in a second? XVII. INVOLUTION. 191. Involution is the process of raising a quantity to any required power. This is effected, as is evident from Art. 13, by taking the quantity as a factor a number of times equal to the exponent of the required power. 192. If the quantity to be involved is positive, all its powers will evidently be positive; but if it is negative, all its even powers will be positive, and all its odd powers negative. Thus, (-α) = (-α) × (-a) × (-a) × (-a) = + a*; etc. X X X Hence, the EVEN powers of any quantity are positive; and the ODD powers of a quantity have the same sign as the quantity itself. INVOLUTION OF MONOMIALS. 193. 1. Find the value of (5a2c)4. (5a2c)* = 5 a2c × 5a2c × 5a2c × 5 a2c = 625 ac, Ans. 2. Find the value of (-3m2)3. (-3m2)3 = (-3m2) × (-3m2) × (-3m2) = -27 m2, Ans. X X From the above examples we derive the following rule : Raise the numerical coefficient to the required power, and multiply the exponent of each letter by the exponent of the required power. Give to every even power the positive sign, and to every odd power the sign of the quantity itself. A fraction is raised to any power by raising both numera tor and denominator to the required power. SQUARE OF A POLYNOMIAL. 194. We find by multiplication : a2+2ab+2ac+b2+2bc + c2 This result, for convenience of enunciation, may be written as follows: (a + b + c)2 = a2 + b2+c2 + 2ab + 2ac+2bc. In a similar manner, we find : (a+b+c+d)2 = a2 + b2 + c2 + d2 and so on. +2ab+2ac+2ad+2bc+2bd+2cd; We have then the following rule for the square of any polynomial: Write the square of each term, together with twice its product by each of the following terms. EXAMPLES. 1. Square 2x2 3x-5. The squares of the terms are 4x, 9x, and 25. Twice the first term into each of the following terms gives the results, 12x and -202; and twice the second term into the following term gives the result, 30x. Hence, 2 (2x2 - 3x - 5)2 = 4x2 + 9 x2 + 25 – 12x3 – 20x2 + 30x = 4x2 - 12x – 11 x2 + 30x + 25, Ans. Square the following expressions : |