first with, and the second against a current of 2} miles per hour. In still water, each would sail 7 miles per hour. In what time will they meet, and what will be the distance of each from its present position ? 205. At 6 per cent. compound interest, what annual payment will extinguish a debt of $3500000 in 50 years ? 206. The Winchester bushel is 181 inches wide and 8 inches deep. What are the dimensions of a tub of similar form that holds 64 bushels ? 207. How much should an annual rent be increased, to amount in 17 years, to the same as $2500 at 6 per cent. compound interest ? 208. At what time between half past 7 and 8 o'clock, are the hour and minute hands exactly 13 minutes apart ? 209. An estate of $4896 is to be divided in such manner that the widow shall receive { as much as the son, and the son ; of 2 times as much as the daughter. What must each receive ? 210. Three men bought a grindstone 50 inches in diameter, for which A. paid 75 cents, B. $1.50, and C. $2.00. What part of the diameter ought each to wear away, allowing the diameter of the axle to be 2 inches? 211. A vessel of 400 tons has a keel 48 feet long. What length of keel has a vessel of 750 tons, that is built on the same model? 212. The commissioners of a certain county are about building a new court house, which will cost $75000. They hire money for the purpose at an annual interest of 5 per cent, and propose to pay the debt thus incurred by 50 equal annual instalments. What amount must be paid each year ? 213. Sold a quantity of sugar for $150, thereby losing 15 per cent., but I ought to have gained as much per cent. as the sugar cost. What was my total loss ? 214. An estate of $20000 is to be divided between two sons in the following manner : the elder is to receive $100 the first month, $300 the second month, &c., in arithmetical progression--and the younger is to receive $1000 per month, until the whole is paid. What is the share of each, and how long will they be in receiving it? annum, 215. What is the quotient of 903 times half a dozen dozen, divided by 121 times six dozen dozen ? 216. A ladder standing upright against a wall reaches the top, but the foot being removed 13 feet from the wall, it reaches to a point 6 feet from the top. Required the length of the ladder, and the height of the wall. 217. There are two cannon balls, one weighing 28 pounds, and the other 9 pounds. What is the diameter of the greater—that of the less being 5 inches ? 218. If one-third of six were three, | What would the fourth of twenty be? 219. A man has a perpetual income of $350 per annum, which he desires to exchange for a life annuity. Supposing money to be worth 6 per cent. a year, what annuity must he receive, it being probable that he will live 16 years? 220. A man hires a farm for 15 years, and expends $1200 on improvements which yield him 7 per cent. per What is the present worth of the gain or loss on his expenditure ? 221. Find the value of x from the equation Ý x = 75. 222. A merchant received on consignment, three bales of sheeting, marked A., B., and C. A. contained 420 yards of a quality 15 per cent. better than B., B. contained 380 yards of a quality 10 per cent. poorer than C., and C. con. tained 450 yards. The whole were sold together at 12} cents per yard; how much should be credited to each, after deducting 2} per cent. commission ? 223. An estate was offered for sale for $12000, but the price appearing too high, the tenant took a lease for 25 years at $800 per annum. How much did he gain or lose, estimating compound interest at 6 per cent. ? 224. Reduce VAX 17 to a continued fraction. 225. A man wishes to give his son, who is now 5 years old, $20000 when he is 21 years old. How much must he invest annually for that purpose, at 6 per cent. compound interest? 226. A farmer has a stack of hay, from which he sells a quantity which is to the quantity remaining, as 4 to 5. He then uses 15 loads and finds that the quantity left, is to the quantity sold, as 1 to 2. Required the number of loads at first in the stack. 227. There are two, numbers, the greater of which is 3 times the less, and the sum of their squares is 8} times the sum of the numbers. What are they ? 228. Find two numbers in the proportion of 4 to 7, the square of whose product is 63504. 229. A farm of 750 acres is divided among three persons. B. has as much as A. and C. wanting 10 acres, and the shares of A. and C. are to each other in the proportion of 7 to 3. Required the share of each. 230. If 11 oxen eat 244 acres of grass in 5 weeks, and 10 oxen eat 19,4 acres in 4 weeks, how many acres of similar pasture will 42 oxen eat in 7 weeks, the grass growing uniformly? 231. What number is that which is 169 greater than the greatest square number below, and 114 less than the least square number above itself? 232. A. and B. can do is of a piece of work in a day, B. and C. can do { of it in 2 days, and A. and C. can do of it in 41 days. In what time would each do it alone, and in what time would it be done if they all worked together? 233. Find the least 3 integers, such that of the first, of the second, and of the third, shall be equal. 234. An estate of $20000 is to be divided among three sons, two daughters, and a widow ; each son is to receive 3 times as much as a daughter, and each daughter twice as much as the widow. Required the share of each. 235. An usurer left his fortune to be disposed of in the following manner: To A. ž, to B. to C. ), to D. ? 27 to E. do, to F. and the remainder, which was $800, was to be paid to C. What was the whole estate, and what was the share of each ? 236. For the lease of a certain estate, A offers $150 premium, and $300 rent per annum; B. offers $400 premium, and $250 per annum; C. offers $650 premium, and $200 3 10 1 50" per annum; and D. offers $1800 in ready money. Whose offer is the best, and what is the difference between them, computing 5 per cent, compound interest? 237. Which is of the greater value, the income of an estate of $500 a year for 15 years to come, or the reversion of the same estate for ever, at the expiration of the 15 years, interest at 6 per cent. ? 238. If a ball were put in motion by a force which would drive it 12 miles the first hour, 10 miles the second, and so on in geometrical progression, what distance would it go in the whole ? 239. What is the least number, which, if divided by 2, will leave 1 remainder, by 3 will leave 2, by 4 will leave 3, by 5 will leave 4, by 6 will leave 5, but by 7, will leave no remainder ? 240. Required the least three numbers, which, divided by 20, will leave 19 remainder, if divided by 19 will leave 18, and so on, (always leaving one less than the divisor,) to unity. 241. A trader offers to receive a young man as partner, proposing, if he will advance $500, to allow him $200 per an. num ; if he will advance $1000, 10 allow $275 per annum ; and if he advance $1500, he will allow $350 per annum. What per cent, is offered for the use of the money, and how much for the young man's time? 242. Arnold's stock in the firm of Arnold and Benton, exceeded Benton's by $2000; Arnold's money continued in trade 5 months, and Benton's 9 months. Required each man's investment, the gain being equally divided. 243. A shepherd sold to one man half his flock and half a sheep, to a second half the remainder and half a sheep, and to a third half the remainder and half a sheep, when he had 20 left. How many had he at first ? END OF PART SECOND, 18* USE OF THE TABLES. I. PRIME AND COMPOSITE NUMBERS. The hundreds are placed at the head of the table, and the tens and units at the left hand. Then, if it be desired to find 5609, look for 56 at the top, and 09 at the side, and at the angle of meeting we find 71, which is one of the factors. Dividing by 71 we obtain 79, which being a prime number is the only remaining factor. For numbers below 1000, all the factors are given. For all odd numbers above 1000, one or more factors will be found in the table, which will reduce the number to a prime or to some composite number less than 1000, when the remaining factors can be found. Primes are indicated by a dash (-). The index written against any number, indicates the number of times it is employed as a factor. Thus, 32 :3 X 3; 53 5 X 5 X 5. II. COMPOUND INTEREST. The amount of any principal, is found by multiplying the principal by the amount of $1.00 for the time. If the time consists of years, months, and days, the amount for the given number of years, must be multiplied by the amount of $1.00 for the remaining inonths and days. The present worth at compound interest, of any sum due at a future time, is found by dividing the given sum by the amount of $1.00 for the time. III. THE AMOUNT OF ANNUITIES. To find the amount of any annuity; multiply the given annuity by the amount of an annuity of $1.00 for the time. If the annuity is payable semi-annually, take the amount for twice the time, at half the rate. IV. PRESENT WORTH OF ANNUITIES. Multiply the given annuity by the present worth of an annuity of $1.00 for the given time. The present worth being given, to find the annuity; divide the present worth by the present worth of an annuity of $1.00 for the same time. |