27. What will be the yearly income from 173 acres, 2 roods 14 poles of land, at the rate of £1 7s. 8d. per acre? Ans. £240 2s. 720d. 28. If 9cwt. of sugar cost £54; what will 7cwt. 3qrs. 14lb. cost? Ans. £47 5s. 29. A, owes B $350, but B compounds with him for 65 cents on the dollar; what must B receive for his debt? Ans. $227,50cts. 30. A bankrupt owes 1250 dollars, and his money and effects amount to $750,50 cents; what is a creditor entitled to, whose demand is 40 dollars? Ans. $24,01ct. 6m. 31. If a person's income be $1877,50cts. a year; how much may he spend each day, to save, at the year's end, $600 ? Ans. $3,50cts. 32. If a man receive 7 dollars interest a year, for the use of 100 dollars; what should he receive for the use of 375 dollars, the same time ? Ans. $26,25cts. 33. If the interest of 375 dollars for one year, be $26,25cts.; what will it be for 19 days? Ans. $1,36cts. 6m. 34. Bought 10 pieces of cloth, each piece containing 25 yards, at $2,50cts. per yard; what did the whole come to? Ans. $643,75cts. 35. Bought 120 gallons of brandy for $187,50cts. ; how much water must be added to the brandy, to reduce the first cost to $1,25cts. ? Ans. 30gal. NOTE. When the given number is in barrels, bales, packages, or pieces, each containing an equal quantity, reduce the quantity contained in one, to its lowest denomination, and then multiply by the given numbers of packages or pieces. If the given number of barrels, bales, pieces, &c. be of unequal content, put the separate content of each under one another, as in Compound Addition, "then add them, and their, sum will be the whole quantity. to? 36 Bought 3 pieces of cloth, containing 36, 341, and 281⁄2 yards, at $2,50 per yard; how much did the whole amount Ans. $247,50cts. 37. Bought 3 hogsheads of rum, containing 601, 621, and 61 gallons, at 90 cents per gallon; what did the whole come to? Ans. $165,60cts. 38. Suppose a freight boat leave Utica for Albany, 12 hours before a packet boat, and go at the rate of 2 miles an *This sum will require two statements; one to ascertain the time, and the other the distance. hour, the packet follows at the rate of 4 miles an hour; in what time, and at what distance will the packet overtake the freight boat? Ans. S The packet will overtake the freight boat, in 20h. and at the distance of 80 miles from Utica. 39. A, leaves the city of New York, to go to Montreal in Lower Canada, and travels at the rate of 35 miles a day; B, at the same time, leaves Montreal to go to N. York, and travels the same road, at the rate of 30 miles a day; how far from the city of New York will they meet, allowing the distance to be 390 miles? Ans. 210 miles. 40. As I was hunting on the forest grounds, Was fourscore and sixteen rods, just, and no more ; How far they ran before they caught the hare? Ans. 336 rods. 41. If 80 gallons of water, in one hour, be conducted into a cistern which will hold 400 gallons, and a conductor convey 30 gallons from the cistern in an hour; in what time will it be filled? Ans. 8 hours. 42. The valuation of the property in a certain town according to the town's inventory, is $610,000, and the tax levied on that town, is $3050; what is A's tax, whose estate according to the inventory is valued, at $1500 ? Ans. $7,50cts. NOTE. As the valuation of the whole property in the town, is to the whole tax, so is the valuation of each man's estate, to his tax. 43. If a tax of $650 be laid on a certain school district, for erecting a school house, and the inventory of all the estates, in that district, amount to $130,000; what must A, pay whose estate is $2500? Ans. $12,50cts. 44. The time made out by the inhabitants of a certain school district, amounts to 3040 days, during the winter, and the teacher is to have $75 for his services; what must A pay, who has sent 80 days to the school? Ans. $1,97cts. 3m. 45. How many yards of carpeting that is half a yard wide, will cover a room 18ft. wide, and 20ft. long? Ans. 80yds. NOTE. To obtain the surface of a plane, as a floor, field &c. Multiply the length, by the breadth, and the product will be the surface; then as the number of square feet in a yard of the carpeting which is 4,5 feet, is to one yard, so is the number of square feet in the floor, to the number of yards required. 46. How many bricks that are 4 inches wide, and 8 inches long, will lay a floor sixteen feet wide, and twenty feet long? Ans. 1440 bricks. 47. If a staff six feet in length cast a shadow of four feet, what is the height of a tree, whose shadow at the same time measures 125 feet? Ans. 187 feet 6 inches. 48. A merchant bought 8 hogsheads of brandy, for 950 dollars; he paid for the freight $100, for the duties and other charges $50; and he wishes to gain $100; how must he sell it per hogshead? Ans. $150. NOTE. It is just the same as if the merchant had, at first, paid out for the brandy the amount of the freight, duties, and other charges, together with the gain and the first cost; and then should wish to know what it cost him per hogshead. 49. A merchant bought 16 hogsheads of wine, at the rate of £87 13s. 6d. per hogshead, New York currency; how many dollars will pay for it, in federal money? Ans. $3507. 50. Suppose a merchant pay 12s. 6d., New England currency, a gallon, for wine; what would 1000 gallons amount to, at that rate, in federal money ? Ans. $2083,33cts. 3m. 51. Bought 300 Ells English, for 500 dollars, but being damaged, I am willing to lose 50 dollars by the sale; how must I sell it per yard? Ans. $1,20cts. 52. If I buy 91b. of coffee for $1,08 cents; how many pounds can I buy for 9 dollars? Ans. 75lb. 53. A gentleman being asked what hour of the forenoon it was, answered, it is between 8 and 9, and the minute and hour hands are together; what was the exact time? Ans. 43 minutes past S. NOTE. The motion of the minute hand, to that of the hour hand, is as 12 to 1, therefore 12-1-11; then as 11 is to 8, so is 12 to the answer. THE RULE OF THREE INVERSE. This Rule, like the Rule of Three Direct, is properly an application of Compound Multiplication and Compound Division, and may like Direct Proportion be reduced back to those rules. This rule is sometimes called Indirect or Inverse Proportion. RULE.-State the questions, and reduce the terms, the same as in the Rule of Three Direct; then multiply the first and second terms together, and divide the product by the third; and the quotient will be the answer, in a denomination of the same name, to which the second term was reduced. In this rule the fourth term or answer bears the same proportion to the second term, as the first term bears to the third. If more require less, or less require more; the question belongs to Inverse Proportion. More requiring less, is when the third term is greater than the first, and requires the fourth term to be less than the second. Less requiring more, is when the third term is less than the first, and requires the fourth term to be greater than the second. NOTE.-The principal difficulty which the student will experience in this rule, will be to distinguish Inverse or Indirect Proportion from Direct Proportion, but as the rule given for stating the questions is the same in both, the student can easily decide to which it belongs, by considering, whether more requires less, or less requires more. EXAMPLES. 1. There was a certain dam erected across a river, in 60 days, by 12 workmen ; but the same being swept away, it is required to rebuild it in 30 days; how many men must be employed about it, working at the same rate? Ans. 24 men. d. men. d. 60 12 30 60 310)720 24 Ans. DEM. From the conditions of this question, it is plain, that it belongs to Inverse Proportion, because it is evident, the less the time, the more men must be employed to finish the work.Here our third term is less than the first, and requires the fourth term or answer to be greater than the second term; and it is evident from the conditions of the question, that the fourth term or answer should be double the second term, because it must require double the number of men to perform the same work in 30 days, that it would in 60 days. And it will be seen, that the fourth term or answer, 24 men, bears the same proportion to the second term, 12 men, that the first term, 60 days, bears to the third term, 30 days; thus, 24: 12 :: 60: 30. To reduce this sum back to Compound Multiplication and Division, divide the first term by the third, and multiply the quotient by the second term; thus, 3|0)6|0d. 2 12 24 The answer as before. 2. If a footman perform a journey in 6 days, when the days are 16 hours long; how many days will he require to go the same journey, when the days are 12 hours long? Ans. 8 days. h. d. h. 166: 12 16 36 6 12)96 Ans. 8 DEM. From this example the learner can plainly see, that the footman travelled 96 hours; then when he travelled 16 hours in a day, it is evident, that it took him a less term of time in days, than when he travelled only 12 hours in a day; and it is plain, that every 12 hours are equal to a day, when he travelled only that time in a day; then as often as 12 is contained in 96, the whole number of hours employed in travelling, so many days he must have been employed in travelling, which proves to be 8 days. Ans. 4 days. 3. If 3 teams can plough a field in 12 days; in what time will 9 teams do it? 4. How many yards of shalloon that is 3 quarters wide will line 18yds. of cloth that is 5qrs. wide? Ans. 30yds. 5. If I lend a friend 200 dollars, for 90 days; how long ought he to lend me 450 dollars, to return the favour? Ans. 40 days. 6. If a board be 8 inches wide; how much in length will make a square foot? Ans. 18 inches. 7. How many yards of paper, 3 quarters wide, will paper a room that is 26 yards round, and 3 yards high? Ans. 104 yards. 8. How much land, at $3,75 cents per acre, should be given in exchange for 300 acres, at $6,50 cents per acre? Ans. 520 acres. 9. How many men must be employed to do a piece of work in five days, which 8 men can do in fifteen days? Ans. 24 men. 10. If 4 men will set 250 apple trees, in six days; how many men will set the same number in two days? EXAMPLES, Ans. 12 men. Promiscuously placed, in Direct and Inverse Proportion. 1. If of a farm be worth 1250 dollars, what is the whole farm worth? Ans. $2250. DEM.-Here it is plain, that the farm is divided into 9 equal shares, or 9 parts; then it is evident, that as 5 shares, are to the worth of 5 shares, that is, $1250, so are 9 s., the whole farm, to the worth of the whole. Shares. $ Shares. 5)11250 Ans. $2250 |