35. A train, after traveling an hour from A towards B, meets with an accident which detains it half an hour; after which it proceeds at four fifths of its usual rate, and arrives an hour and a quarter late. If the accident had happened 30 miles farther on, the train would have been only an hour late. Find the usual rate of the train. and Let Then, Then, y 5x = the number of miles from A to B, the number of miles the train travels per hour. 4x the rate of the train after the accident. y5x the number of miles the train has to go after = the accident. the number of hours required usually, the number of hours actually required. But since the train was late, the running time was = the loss in hours of running time. y-5x y-5x = 3 If the accident had happened 30 miles farther on, the remainder of the journey would have been running time would have been From the solution of equations (1) and (2), x = 6, and 5x = 30. Therefore, the usual rate of the train is 30 miles an hour. 36. An express train, after traveling an hour from A towards B, meets with an accident which delays it 15 minutes. It afterwards proceeds at two thirds its usual rate, and arrives 24 minutes late. If the accident had happened 5 miles farther on, the train would have been only 21 minutes late. Find the usual rate of the train. 37. If 3 yards of velvet and 12 yards of silk cost $60, and 4 yards of velvet and 5 yards of silk cost $58, what is the price of a yard of velvet and of a yard of silk? 38. If 5 bushels of wheat, 4 of rye, and 3 of oats are sold for $9; 3 bushels of wheat, 5 of rye, and 6 of oats for $8.75; and 2 bushels of wheat, 3 of rye, and 9 of oats for $7.25; what is the price per bushel of each kind of grain? 39. A train proceeded a certain distance at a uniform rate. If the speed had been 6 miles an hour more, the time occupied would have been 5 hours less; but if the speed had been 6 miles an hour less, the time occupied would have been 7 hours more. Find the distance. HINT. If x = the number of hours the train travels, and y the number of miles per hour, then xy: the distance. 40. A certain number of persons paid a bill. If there had been 10 persons more, each would have paid $2 less; but if there had been 5 persons less, each would have paid $2.50 more. Find the number of persons and the amount of the bill. 41. A man bought 10 cows and 50 sheep for $750. He sold the cows at a profit of 10 per cent, and the sheep at a profit of 30 per cent, and received in all $875. Find the average cost of a cow and of a sheep. 42. It is 40 miles from Dover to Portland. A sets out from Dover, and B from Portland, at 7 o'clock A.M., to meet each other. A walks at the rate of 3 miles an hour, but stops 1 hour on the way; B walks at the rate of 21 miles an hour. At what time of day and how far from Portland will they meet? 43. A number is expressed by three digits. The sum of the digits is 21; the sum of the first and second exceeds the third by 3; and if 198 is added to the number, the digits in the units' and hundreds' places will be interchanged. Find the number. 44. If the length of a rectangular field is increased by 5 yards and its breadth by 10 yards, its area is increased by 450 square yards; but if its length is increased by 5 yards and its breadth diminished by 10 yards, its area is diminished by 350 square yards. Find its dimensions. 45. If the floor of a certain hall had been 2 feet longer and 4 feet wider, it would have contained 528 square feet more; but if the length and width were each 2 feet less, it would contain 316 square feet less. Find its dimensions. 46. If the length of a rectangle was 4 feet less and the width 3 feet more, the figure would be a square of the same area as the given rectangle. Find the dimensions of the rectangle. 47. If a certain number is divided by the sum of its two digits diminished by 2, the quotient is 5 and the remainder 1; if the digits are interchanged, and the resulting number is divided by the sum of the digits increased by 2, the quotient is 5 and the remainder 8. Find the number. 48. A person has a certain capital invested at a certain rate per cent. Another person has $2000 more capital invested at one per cent better than the first, and receives $150 more income. A third person has $3000 more capital invested at two per cent better than the first, and receives $280 more income. Find the capital of each, and the rate at which it is invested. 49. A man makes an investment at 4 per cent, and a second investment at 4 per cent. His income from the two investments is $715. If the first investment had been made at 4 per cent and the second at 4 per cent, his income would have been $15 greater. Find the amount of each investment. 50. A number is expressed by two digits, the units' digit being the larger. If the number is divided by the sum of its digits, the quotient is 4. If the digits are reversed and the resulting number is divided by 2 more than the difference of the digits, the quotient is 14. Find the number. 51. An income of $335 a year is obtained from two investments, one in 4 per cent stock and the other in 5 per cent stock. If the 4 per cent stock should be sold at 110, and the 5 per cent at 125, the sum realized from both stocks together would be $8300. How much of each stock is there? 52. A sum of money, at simple interest, amounted in m years to c dollars, and in n years to d dollars. Find the sum and the rate of interest. 53. A sum of money, at simple interest, amounted in m months to a dollars, and in n months to b dollars. Find the sum and the rate of interest. 54. A person has $18,375 to invest. He can buy 3 per cent bonds at 75, and 5 per cent bonds at 120. How much of his money must he invest in each kind of bonds in order to have the same income from each investment? HINT. Notice that the 3 per cent bonds at 75 pay 4 per cent on the money invested, and 5 per cent bonds at 120 pay 4 per cent. 55. In a mile race A gives B a start of 44 yards, and is beaten by 1 second. In a second trial A gives B a start of 6 seconds, and beats him by 93 yards. Find the number of yards each runs a second. It 56. A train, after running 2 hours from A towards B, meets with an accident which delays it 20 minutes. afterwards proceeds at four fifths its usual rate, and arrives 1 hour 40 minutes late. If the accident had happened 40 miles nearer A, the train would have been 2 hours late. Find the usual rate of the train. 57. A boy bought some apples at 3 for 5 cents, and some at 4 for 5 cents, paying $1 for the whole. He sold them at 2 cents apiece, and cleared 40 cents. How many of each kind did he buy? 58. Find the area of a rectangular floor, such that if 3 feet were taken from the length and 3 feet added to the breadth, its area would be increased by 6 square feet, but if 5 feet were taken from the breadth and 3 feet added to the length, its area would be diminished by 90 square feet. 59. A courier was sent from A to B, a distance of 147 miles. After 7 hours, a second courier was sent from A, who overtook the first just as he was entering B. The time required by the first to travel 17 miles added to the time required by the second to travel 76 miles is 9 hours 40 minutes. How many miles did each travel per hour? 60. A box contains a mixture of 6 quarts of oats and 9 of corn, and another box contains a mixture of 6 quarts of oats and 2 of corn. How many quarts must be taken from each box in order to have a mixture of 7 quarts, half oats and half corn? 61. A train traveling 30 miles an hour takes 21 minutes longer to go from A to B than a train which travels 36 miles an hour. Find the distance from A to B. 62. A man buys 570 oranges, some at 16 for 25 cents, and the rest at 18 for 25 cents. He sells them all at the rate of 15 for 25 cents, and gains 75 cents. each kind does he buy? How many of 63. A and B run a mile race. In the first heat B receives 12 seconds start, and is beaten by 44 yards. In the second heat B receives 165 yards start, and arrives at the winning post 10 seconds before A. Find the time in which each can run a mile. |