51. A gentleman hired a mason to build a cellar wall which contained 150 cords of rough stone, and paid for the stone and building, at the rate of 64 cents per solid foot; how many solid feet did the wall contain, and how much did the stone and the laying of it cost him? No. of feet, 4500. Ans. Amount, $281,25, 52. Bought 25 pipes of wine, each containing 128 gallons, at 256 dollars a pipe; how many quarter casks of 314 gallons each will contain the whole, and how must I sell it per quarter cask, to make 1600 dollars by the speculation? S 101 casks, 18 gallons over. Ans. Must sell it for $79,20,7 per qr. cask. 53. A man bought 15 casks of cider, each 126 gallons; how many barrels, each 31 gallons, and how much did he pay for it at the rate of 2 dollars, 37 cents a barrel? No. of barrels, 60. Ans. Amount, 142 dolls. 50 cts. 54. Bought 3 pipes of wine, containing 125, 124, and 122 gallons, for 512 dollars; what did it cost a gallon, and how must I sell it a gallon to gain $92,75 on the whole ? Ans. Cost $1,38 a gal. I must sell at $1,63 a gal. 55. If a gentleman's income be 1836 dollars a year, and he spends 3 dollars, 49 cents a day, how much will he have saved at the year's end? Ans. 562 dollars, 15 cents. 56. Bought a bale of cloth, containing 240 yards, for 600 dollars; how must I sell it per yard to gain 120 dollars on the whole ? 57. Bought a pipe of wine for 327 paid for it at the rate of 2 dollars, 37 many gallons did the pipe contain? Ans. 3 dollars. dollars, 75 cents, and cents a gallon; how Ans. 138 galls. 58. A merchant in Providence, remitted to his correspondent in New-York, 1450 dollars, requesting him to purchase in equal quantities, Hyson tea, at 87 cents a pound; Young Hyson at 62 cents a pound, and Souchong at 50 cents; how many pounds of each kind can he purchase with the money? how many lbs. in all; how many chests, each weighing 62 lbs.? Ans. 725 lbs. of each: 2175 lbs. in all: and 35 chests. 59. If a man drink 3 glasses of rum every day, at the cost of 3 cents a glass, how much will he drink in 20 years; how much will it cost him in that time; how many acres of land, at 12 dollars an acre, could he purchase for the money ? Ans. He will drink 342 galls. 0 qts. 1 pt. 2 gills. It will cost him $657, and would purchase 54 acres of land. 60. How many yards of carpeting, that is yards wide, will cover the floor of a room, that is 5yards long, and 4 yards in width? Ans. 263 yards. 61. How much time would a man redeem, by rising one hour earlier every day, than his usual custom, if he continued from the age of 21, to that of 60 years; and allowing his time to be worth 12 cents an hour, how much would he add to his fortune by so doing? Ans. He would redeem 593 days, 3 hours. And would add to his fortune, $1779,37. 62. A farmer harvested 25 acres of corn that yielded 46 bushels to the acre; what is the whole worth at 87 cents the bushel? and allowing 2150 cubic inches to the bushel, how many cubic feet must a bin contain to hold it all? Ans. { Amount, 1006 dollars, 25 cents. No. of feet in the bin, 1430: 1460 inches. 63. A merchant bought 75000 hoops at $18,50 a thousand; 120000 shingles at $4,25 a thousand; and 4650 staves at $2,37 a hundred; and paid in part, with 575 gallons of mo lasses at 37 cents a gallon, and the balance in cash; how much money was due on the balance. Ans. $1792,301. 64. If 4 men can row a boat 10 miles an hour in still water, how long will they be in going from New-York to Albany, a distance of 160 miles, allowing the current to retard them 2 miles an hour, and the wind and tide to advance them half a mile an hour? Ans. 18 h. 49 m. 2412 sec. RULE OF THREE. Q. On what established principle or law of Arithmetic is the Rule of Three founded? A. On the principle or law of proportion. Q. What is to be understood by the law of proportion? A. That all like quantities, whether known or unknown, bear a certain proportion to each other. Q. What then are we taught by this rule? A. We are taught by having certain quantities given, bearing a known proportion to each other, to find other unknown quantities, which shall bear a like proportion to some of the given quantities. Q. How many terms are given in the RULE OF THREE? A. Three terms are always given, to find a fourth, which will bear the same proportion to one of the given terms, as the other two bear to each other. Q. How many kinds of Rule of Three are there? A. Two: Rule of Three Direct, and Rule of Three Inverse. Q. What proportion do the terms bear to each other, in the Rule of Three Direct? A. The fourth term, or answer sought, will bear the same proportion to the third term given, as the second bears to the first. Q. What proportion do the terms bear to each other, in the Rule of Three Inverse? A. The fourth term, or answer sought, bears the same proportion to the second term given, as the first term bears to the third. Q. How are we taught to answer questions by this rule? A. By stating the question, and saying, as the first term: is to the second so is the third term to the fourth or answer. Q. How is the first term distinguished from the other giv en terms? A. The first term is always that which bought, cost or produced one of the other terms. Q. How is the second term distinguished? A. It is that which was caused, bought or produced by the first term, and is always of the same name or kind as the fourth term or answer sought. Q. How is the third term known? A. It is always of the same name or kind with the first; and is that which must buy, cause or produce the fourth term The third term is also a demand, and may be known by these or the like words, viz: What will, what cost, how much, how far, how long, &c. before it. or answer. Q. How then do you state questions in this rule? A. Say If the 1st term bought, what the 2d 3d term will term the $9 { buy, cost, or produce. Q. If the terms consist of several denominations, how do you proceed? A. Reduce the FIRST and THIRD terms to the lowest denomination mentioned in either: then reduce the SECOND term to the lowest denomination mentioned in that term; and if the proportion be direct, multiply the second and third terms together, and divide their product by the first term, and the quotient will be the fourth term or answer, in the same denomination you left the second term in, and must be brought to the denomination required by the question. Q. If the proportion be Inverse, how do you proceed? A. Multiply the first and second terms together; divide the product by the third term, and the quotient will be the answer in the same denomination in which you left the second term, and must be brought to the denomination required by the question. Q. How do you distinguish the Rule of Three Direct from the Rule of Three Inverse? A. If the third term be greater than the first, and require the fourth term or answer to be greater than the second; or if the third term be less than the first, and require the fourth term to be less than the second, the question belongs to the Rule of Three Direct. Q. How can you distinguish the Rule of Three Inverse? A. If the third term be greater than the first, and require the fourth term or answer to be less than the second; or if the third term be less than the first, and require the fourth term to be greater than the second, the question belongs to the Rule of Three Inverse. NOTE. Although this rule is seldom used in business, and is, by many modern authors left out of their works, still, as so large a proportion of arithmetical calculations are founded upon its principles, and as system in the order of arrangement is a prime object in this work, I have thought best to give it a place in the order it has so long maintained in productions of this kind. EXAMPLES FOR MENTAL EXERCISE, SERIES 1. 1. If 1 yard of cloth cost 4 dollars, what will 6 yards cost? 2. If 8 yards cost 48 dollars, what will 1 yard cost? 3. If 6 yards cost 3 dollars, what will 8 yards cost? 4. If 9 yards cost 6 dollars, what will 12 yards cost? 5. What number will bear the same proportion to 4, that 3 bears to 2? 6. If a man can travel 15 miles in 3 hours, how many miles can he travel in 4 hours? 7. If 2 men can do a job of work in 6 days, in how many days can 4 men do the same? 8. If 7 inen can build a house in 6 weeks, in how many weeks can 6 men build the same? 9. If an ell French of lace cost 6 dollars, what will 1 yard of the same cost? 10. If 1 yard of cloth cost 8 dollars, what must I give for an 'ell Flemish of the same? 11. If 7 bushels of corn cost $33, what will 9 bushels cost? 12. If 5 sheep cost 7 dollars, what will 8 sheep cost? 13. If 3 men can mow a meadow in 3 days, how many days will 9 men be in mowing the same? 14. How many bushels of oats, at 25 cents a bushel, must I give, for 5 bushels of corn, at 75 cents a bushel? 15. A. can walk 30 miles in a day, and B. can walk 35 miles in the same time; now if A. set out 2 days before B. how many days before B. will overtake A.? 16. If 3 engines can empty a reservoir in 60 minutes, how long will 5 engines be in emptying the same? SECOND SERIES OF MENTAL EXAMPLES. 1. What number will bear the same proportion to 7, that 14 bears to 8? 2. What number will bear the same proportion to 9, that 3 bears to 4? What inversely? 3. As 5 is to 9, so is 6 to what number? 4. As 3 is to 1, so is 8 to what number? 5. As 2 is to 7, so is 6 to what number? 6. As 1 is to 9, so is 5 to what number? 7. As 2 is to 12, so is 4 to what number? 8. As 3 is to 15, so is 2 to what number? 9. As 4 is to 5, so is 7 to what number? 10. As 6 is to 8, so is 3 to what number? 11. As 7 is to 10, so is 1 to what number? What inversely? 12. As 8 is to 11, so is 9 to what number? What inversely? 13. As 9 is to 12, so is 4 to what number? What inversely? 14. As 10 is to 3, so is 11 to what number? What inversely? 15. As 11 is to 2, so is 10 to what number? What inversely? 16. As 12 is to 4, so is 11 to what number? What inversely? 17. If wide require 4 yards, what will wide require? 18. If wide require 2 yards, what will wide require? 19. If 5 men can earn 8 dollars in 1 day, what will 7 men earn in the same time? 20. Five men, in dividing a sum of money, had each 7 dollars; now if the same sum had been divided among 7 men, how many dollars would each have had? 21. If a hare start 40 yards before a hound, and the hound gain 6 yards a minute, how long will the chase last? 22. A certain quantity of provision will last 5 men 3 months; how long will it last 8 men? 23. If 5 men will eat 7 pounds of meat in a given time, how many pounds will 8 men eat in the same time? 24. A cistern. has 2 faucets, either of which will empty it in 45 minutes; if both be set running together, in what time will they empty it? 25. A cistern has 2 faucets, one of which will fill it in 30 minutes, the other will empty it in 60 minutes; if both be set running, in what time will it be filled? |