39. If a draper sell coarse cloth at 11s. 6d. per yard, and thereby gain 15 per cent. what would he gain per cent. by selling the same cloth at 12s. per yard ? Ans. 20 per cent. 40. If by selling corn at 8s. per bushel, the merchant gains 10 per cent ; but the market falling, he was obliged to sell the rest at 7s. per bushel; what did he gain or lose on this last sale ? Ans. he lost 3} per cent. 41. The rents of a parish amount to £3500, and a rate is granted of £131 58. What is that per pound ? Ans. . 9d. 42. How much must be paid for the carriage of 10-cwt. 2 qrs. at the rate of 24d. per lb ? Ans. £11. Os. 6d. 43. If a family of 7 persons consume 3 bushels of meal in a month, how many bushels will serve them when they are 21 in family? Ans. 9 bushels. 44. Some sportmen having placed a fox 100 yards distant from two hounds, and starting both together, the hounds ran 21 times faster than the fox. I demand how far the fox will have run before the hounds overtakes her ? Ans. 66 yds. 45. A person being asked what hour of the afternoon it was, answered, it is between 5 and 6, and the minute and hour hands are together. Required the exact time ? 'Ans. 274 m. past 5. 46. A dealer bought 14 pipes of wine, and is allowed 6 months credit, but for ready money gets it 6d. in a gallon cheaper. How much did he save by paying ready money? Ans. £44 2s. 47. The earth revolves round the sun in 365 days, 6 hours, 9 minutes, describing according to Dr. Stewart's theory, an orbiť of 744457824 miles. How many miles does it move in a minute ? Ans. 141521198 miles. 48. If 12 pears are worth 21 apples, and 3 apples cost a cent, what will be the price of fourscore and four pears? Ans. 49 cents. 49. If a lever be 100 inches long, what weight lying 7} in. ches from the end, resting on a pavement, may be moved with the force of 168 lb. fifting at the other end of the lever ? Ans. 2072 lbs. 50. Bought threescore barrels of provisions, for 3 times as many pounds, and sold them again for 4 times as much ; but if they had cost me as much as I sold them for, what should I have sold them for, to gain after the same rate ? Ans. £320. 51. If the distance between two places be 197 miles, how many times will a wheel of 144 feet circumference turn round in performing the journey! Ans. 72993.1 , times. 52. If during ebblat tom, a wherry should set out from Albany to go down the her, and at the same time another should set out from Hudson, to come up the river, allowing the distance to be 30 miles : Suppose the current forwards one and retards the other 21 miles an hour ; the boats are equally laden, the rowers equally good, and in the ordinary way of working in still water, would proceed at the rate of 5 miles an hour. The question is where in the river will the two boats meet? Ans. $ 22 m. from Albany. 7 m. from Hudson. 53. The swiftest velocity of a canron ball is about 2000 feet in a second of time ; then in what time, at that rate, would such a ball be in moving from the earth to the sun ; admitting the distance to be 100 million of miles, and the year to contain 365 days, 6 hours ? Ans. 501 yrs. 49 w. 2 ho. 54. The slow or parade step being 70 paces per minute, at 28 inches per pace, it is required to determine at what rate per hour that movement is ? Ans. 1 mile, 6 fur. 33 per. 5 yds. O ft. 6 in. 55. If in 4 months I spend as much as I gain in 3 months, how much can I lay up at the years end, if I gain every 6 months £150 ? Ans. £75. 56. If a piece of velvet 30 yards long, cost £15, how much will a Dutch ell come to if 3 yards are equal to 4 ells ? Ans. 73. 6d. 57. If 2 lb. of pepper cost half a dollar, what will 60 lb. of cloves come to, if 3 lb. of cloves are worth 16lb. of pep Ans. $80. 58. A. travels 12 miles a day, and when he has been gone 15 days, B. sets out after him. I demand how many miles in a day B. must go to overtake A. in 60 days? Ans. 15 miles. 59. If by selling tea at 138. 6d. a lb. I gain 35 per cent. Pray what was it bought for ? Ans. 10s. 60. If 10 and 6 make 20, how much makes 6 and 8 ? Ans. 61. Suppose a grey-hound makes 27 springs whilst a hare makes 25, and the springs are alike : now if the hare is 50 springs before the hound, how many springs will the hound make in overtaking the hare? Ans. 675 62. When I sell a yard of Oznaburgh at 15d. I gain 30s. by the piece, and when I sell the yard at 14d. I gain 20s. by the same piece. I demand how many yards the piece contained ? Ans. 120. 63. There is a certain cylindrical rod 36 inches long, at the extremities of which are suspended two weights, the one 5 pounds and the other 7. I demand the F. jnt of the rod where these two weights will be in equilih se'ns. 15 in. from 7 16. per ? 17 64. Sold a quantity of muplin for $588, and by so doing lost 16 per cent. whereas in dealing I ought to have cleared 25 per cent. How much then was it sold under the just value? Ans. $287. THE RULE OF THREE INVERSE. Whereas in the Rule of Three Direct, more required more, or less required less, so here the contrary takes place, more requires less, and less requires more. RULE.-As in the Rule of Three Direct, state and prepare the question; then multiply the first and second terms together, and divide the product by the third. The quotient will be the aliswer in the name, or denomination of the se. cond. EXAMPLES. 1. A captain of a ship, being provided with provisions for 40 men, 10 months. How long would those provisions serve 32 men ? Ans. 121 months. 2. A merchant has agreed, with a carrier to transport 12 cwt. of goods 70 miles, for 13 French crowns, but the waggons being laden, the carrier is obliged to leave 2 cwt. how. ever, the merchant will still give the 13 crowns, on condition of carrying the goods proportionably further. Required the number of miles they must be carried ? Ans. 84 miles. 3. If 40 poles in length and 4 in readth, make an acre, what must loe the length to make an acre when the breadth is 15 poles ? Ans. 10 poles, 3 yds. 2 ft. 4. How many yards of paper that is 3 quarters wide will be sufficient to line a room that is 24 yards round and 4 yards high? Ans. 128 yards. 5. If 50 horses are maintained a year in grain for a certain sum, when oats are sold at 28. 3d. per bushel, how many hora ses may be maintained a year in grain for the same sum, when oats are at 2s.6d. per bushel ? Ans. 45 horses. 6. If 136 masons are able to build a fort in 28 days, but that it was required to have it finished in 8 days; what number of masons must be employed se as to complete it in that time? Ans. 476 masons. 7. The governor of a besieged place having provisions for 54 days, at the rate of 13 lb. of bread, but being desirous to prolong the siege to 80 days, in expectation of succour, in that case what must the ration of bread be? Ans. 100 lb 8. Suppose 4000 soldiers in a town, were provided with provisions for 3 months, how many men must the command. ing officer dismiss that the provisions may last 8 months ? Ans. 2500 men a square foot ? 9. How much in length, that is 4 inches broad, will make Ans. 36 inches. 10. A board is 48 inches long, how much in breadth must there be to make 14 square foot ? Ans. 54 inches. 11. A. borrowed of B. $400 for 3 years ; how long ought A. to lend B. $748, to requite his kindness. Ans. 1yr. 7 mo. 744 2 days. 12. There is a cistern having a pipe, which will empty it in 15 hours ; how many pipes of the same capacity will empty it in 3 quarters of an hour ? Ans. 20 pipes. 13. If when wheat is 13s. 6d. per bushel, the shilling loaf weigh 2 lb: 12 oz. what must it weigh when wheat is 10s. per bushel ? Ans. 3 16.11% oz. 14. If 6 men in 24 days mow 72 acres, in how many days will 8 men mow the same ? Ans. 18 days. 15. If a field will feed 6 cows 90 days, how long will it feed 20 cows? Ans. 27 days. 16. A wall that is to be built to the height of 27 feet, was raised 9 feet high by 12 men in 6 days; then how many men must be employed to finish the wall in 4 days, at the same rate of working? Ans. 36 men. 17. Wanting just an acre of land cut off from a piece which is 13} poles in breadth, what length must the piece be ? Ans. 11 poles, 4 yards, 2 feet, czy in. 18. If a horseman rides 100 miles in 33 hours 20 minutes, in what time would be ride the same distance, if he went at the rate of 54 miles an hour ? Ans. 19 ho. 221 mi. 19. If 20 men can perform a piece of work in 12 days, how many men will accomplish another piece thrice as large in onefifth of the time? Ans. 300 men. 20. The battering ram of Vespasian weighed, surpose 100000 lbs. and was moved, let us admit, with such a velo. city by strength of hands, as to pass through 20 feet in one second of time, and this was found sufficient to demolish the walls of Jerusalem ; with what velocity must a bullet that weighs but 30 lbs. be moved, in order to do the same execution ? Ans. 666663 feet per second. 21. If 3 men or 4 women, will do a piece of work in 56 days, how long will one man and one woman be in doing the same ? Ans. 96 days. 22. What weight will a person be able to keep in equilibro who presses with a force of 120 lbs. on the end of an equipoised leaver 60 inches long, which is to meet with a convenient fulcrum or prop on which it is moveable, exactly 12 in. ches above the other end of the machine ? Ans, 480 lbs. The American Tutor's Guide, &c. PART II. VULGAR FRACTIONS. unit,' and consists in two numbers, one wrote over the other with a line between them, as 1, z 2&c. The number below the line is called the denominator of the fraction, because it denominates how many parts the unit is divided into, and the number above the line is called the numerator, because it numerates how many of those parts are contained in the fraction. A vulgar fraction is either proper, improper, compound, or mixed. A proper fraction is when the numerator is less than the denominator, as }, š, 141, &c. An improper fraction is such whose numerator is equal to or greater than its denominator, as j, 185, &c. A compound fraction is the fraction of a fraction, and is known by the preposition of, between them, as, į of of, &c. A mixed number is composed of a whole number and fraction, as 41, 123, 1451%, &c. I. To reduce or abbreviate Fractions to their lowest terme. RULE. Divide the greater term by the lesser, and that divisor by the remainder following, and so on till nothing remains ; then by the last divisor divide both parts of the frae. tion, and the quotients will give the fraction required. EXAMPLES. 1. Reduce 987 to its lowest terms ? Ans. 2. Reduce 252 to its lowest terms ? 3. Reduce 1344 to its lowest terms? 4. Reduce 42000 to its lowest terms? II. To reduce a mixed number to an improper Fraction. RUI.E.-Multiply the whole number by the denominator of the fraction, and to the product add the numerator ; under which set the denominator. A whole number may be expressed like a fraction by wri. ting 1 for its denominator ; thus, 5 is . 3969 47 189 364 1536 5 4 000 |