44. The Dr. side of an account is $800, due May 16, and the Cr. side is $600, due May 1. When can the balance be paid without loss of interest to either party? 45. The Dr. side of an account is $800, due May 1, and the Cr. side is $600, due May 16. When can the balance be paid without loss of interest to either party? men. 46. A detachment of four regiments consisted of 4600 A.'s regiment contained 33 more than B.'s, 95 more than C.'s, and 200 more than D's. How many were there in each regiment? 47. B. is half as old again as A., C. is twice as old as B., and the sum of their ages is 55. Required the age of each. 48. B. is 2 years older than A., C.'s age is 4 years more than the sum of A's and B.'s, and D.'s age, which is 48, is equal to the sum of the other three. What is the age of each? 7, and and, 49. A teacher being asked the number of his pupils, replied, "If I had as many more, and 2, and, 5 as many, I should have 31 less than 200," 10 had he? How many 50. What sum of money will amount to $1500, in 15 years, at 5 per cent. simple interest? 51. At what rate per cent., simple interest, will $700 amount to $1300 in 11 years? 52. In what time will $1100 amount to $1750, at 6 per cent. simple interest? 53. A person, after spending and of his money, had $263 left. How much had he at first? 54. A cistern has three pipes; the first can fill it in an hour, the second can fill it in † of an hour, and the third can empty it in an hour. In what time will the cistern be filled, if they all run together? 55. Five-sevenths of a certain number exceeds of of it by 46. What is the number? 56. A gentleman has a carriage worth $500, and two valuable horses. If the first horse be harnessed in the carriage, the horse and carriage will be worth 3 times as much as the second horse. But the second horse and carriage are together worth 7 times as much as the first horse. What is the value of each? 57. If 2 tons of merchandise can be carried of 961 miles for $3.75, what should be the freight of 1 cwt. for 7 miles less than 72§ miles? 58. A hare starts 5 rods before a greyhound, and runs at the rate of 12 miles an hour. After running 48 seconds, the hound starts in pursuit, and runs 20 miles an hour. In what time will the hare be overtaken? 59. A bathing-tub that holds 147 gallons, is filled by a pipe that brings 14 gallons in 9 minutes, and emptied by a pipe that discharges 40 gallons in 31 minutes. Both pipes having been left open for 3 hours, it is required to find in what time the tub will be filled if the discharging pipe is closed? 60. A spendthrift, after squandering of his fortune, and of the remainder, had $1500 left. How much had he at first? 61. A younger brother received $1100, which was & of his elder brother's fortune, and 5 times the share of the elder brother was twice as much as their father was worth. What was the father worth? 62. Find two numbers in the proportion of 7 to 3, whose product is 189. (Let 7x and 3x be the numbers, then the product will be 21x2.) 63. There is a bin 4 feet high, that contains 299 cubic feet. What is the length of one side, the bottom being square? 64. The length of a certain lot is to the breadth as 11 to 3, and the area is 2673 feet. What are the dimensions of the lot? 65. The product of two numbers is 398, and the quotient of the greater by the less is 23. Required the numbers. 66. The length of a box is to the breadth as 4 to 3; and the depth to the breadth, as 2 to 3. What are the dimen sions, the solid contents being 319144 cubic inches ? 67. Four men joined in a speculation, and the gain was so divided that A., B., and C., together, received $487; A., B., and D., $494; B., C., and D., $506; and A., C., and $481. What was each man's share of the gain? D., 68. Three men traded in partnership. A. contributed $1500, B. $2250, and C. the remainder. The whole gain was $2700, of which C. received $1200. How much did C. contribute, and what did A. and B. gain? 69. A. and B. can do 1 of a piece of work in 1 day; B. and C. can do % of it; A. and C. can do of it in the same time. In what time will they ail do it working together? 70. An estate of $15000 is to be divided among three persons; A. is to receive $5 as often as B. receives $4, and B. is to receive $33 as often as D. receives $44. What is the share of each? 71. Three partners gained $1230, of which A. received $400, B. $350, and C. $480. A's stock was in trade 6 months; B's, 5 months; and C's stock, which was $1800, was in trade 8 months. Required the stock of A. and B. 72. The difference between two numbers is 8, and the sum of their squares is 82. What are the numbers? (Let x = the less, and x + 8 = the greater. The equation will be 2x2+16x+ 64 82. As 64 is added to 2x2 + 16x to make 82, if 64 be subtracted from 82, the remainder must be equal to 2x2+16x. The value of x may then be obtained by the general rule for the extraction of roots.) 73. The difference of two numbers is 11, and their product is 20. Required the numbers. 74. The area of a certain field is 187 square rods, and the length exceeds the breadth by 6 rods. What are the dimensions? 75. There are two numbers whose difference is 4, and their product is to the sum of their squares, as 6 to 13. What are the numbers? 76. A farmer purchased a number of geese for £6 5s. He retained 5, and sold the remainder for 1s. 3d. apiece more than he paid, thus receiving what he paid for the whole. How many did he buy? (Let x = the number, then 1500d. will be the price paid for each, in pence.). CHAPTER XXV. MISCELLANEOUS PROBLEMS. CHRONOLOGY. ACCORDING to the Julian Calendar or OLD STYLE, the solar year was considered as being 365 days and 6 hours. The 6 hours in 4 years amounted to a day, therefore every fourth year was called a Leap Year, and consisted of 366 days. But the true solar year is about 11 minutes less than the Julian year, and on this account, in 1582, it was found that Spring commenced 10 days later than at the establishment of the Julian Calendar. Pope Gregory the XIIIth, therefore, caused ten days to be taken out of the month of Oc tober in that year, and to prevent the recurrence of a similar variation, he ordered the centurial years should not be regarded as Leap Years, unless the number of centuries were divisible by 4. This computation, which is called the Gregorian or New STYLE, was soon adopted in the greater part of Europe; but in England and America, the change was not made until 1752, when the error had amounted to eleven days. It was then ordered that the 3d of September should be called the 14th, and the Gregorian calendar adopted for the future. In Russia and Greece, the Old Style is still retained. One of the first seven letters, A, B, C, D, E, F, G, is attached to every day in the year; thus, A is applied to Jan. 1st, 8th, 15th, &c.; B, to Jan. 2d, 9th, 16th, &c.; C, to Jan. 3d, 10th, 17th, &c. In this manner all days in any year which have the same letter, fall on the same day of the week. The DOMINICAL LETTER for any year is the letter that falls against all the Sundays. Thus, the 5th of January, 1845, will fall on Sunday, and the dominical letter will, therefore, be the 5th letter, or E. But in Leap Year there are two dominical letters, the first for January and February, the second for the remainder of the year. The following are the dominical letters for a few years to come; 1844, G, F; 1845, E; 1846, D; 1847, C; 1848, B, A; 1849, G; 1850, F; 1851, E; 1852, D, C, &c. PROBLEM I. To find the Dominical Letter for any year, according to the Julian or OLD STYLE. RULE. To the given year add one-fourth of itself, plus 4, and divide the sum by 7. If there is no remainder, the dominical letter is G; if 1 remainder, F; and so on in inverse order. If the given year be Leap Year, the letter thus found will be the dominical letter for the last 10 months, and the next following letter, for the remainder of the year. What was the dominical letter for A. D. 1531 ? What were the dominical letters for A. D. 564? 564 141 4 7)709 101 + 2 The remainder 2, indicates that the dominical letter is the 2d from G, or E. But the year being leap year the dominical letter for January and February, will be the next following, or F. The two letters sought are therefore F, E. If the given year were before the Christian era, the remainder would indicate the direct order of the letter. Thus, 1 denotes A; 2 denotes B; 5, E, &c. 1. What was the dominical letter for A. D. 769? 2. What were the dominical letters for A. D. 1492? |