rel? 41. A man had 4 sixths of a week's board for 3 dollars; how much is that for a whole week 1 42. 3 is 5 sixths of what number 2 -- 43. 9 is 5 eighths of what number? > - * 44. 10 is 7 thirds of what number 1 on 45. 11 is 6 fifths of what number 1 of | 46. 12 is 7 fifths of what number 2 : 47. 15 is 2 sevenths of what number 3 48. 17 is 3 tenths of what number 7 49. A man bought 5 eighths of a cask of wine for 19 dollars, what would the whole cask cost 7 50. 19 is 3 eighths of what number 1 51. 21 is 4 fifths of what number 7 52. Bought 5 ninths of a ton of logwood for 23 dollars ; what would a ton cost at that rate 1 53. 23 is 7 ninths of what number 7 : 54. 21 is 3 tenths of what number 7 55. 29 is 4 sevenths of what number? 56. 31 is 5 eighths of what number 1 57. 33 is 6 sevenths of what number 7 58. 38 is 4 ninths of what number 7 59. A man bought 1 barrel and 1 seventh of a barrel of wine for 41 dollars; what was that a bar Note. 1 barrel and 1 seventh of a barrel is the same as 8 sevenths of a barrel. If 8 sevenths of a barrel cost 41 dollars, what does a barrel cost 7 60. 35 is 8 sevenths of what number 3 61. A man bought 1 ton and 4 fifths of a ton (that is, 9 fifths of a ton) of fustic for 43 dollars, what was that a ton 2 62. 52 is 9 fifths of what number 2 63. Bought 1 ton and 2 sevenths of a ton of logwood for 48 dollars, what was that a ton? 64. 67 is 9 sevenths of what number 3 65. 53 is 7 tenths of what number 7 65. 58 is 9 eighths of what number? 67, 61 is 10 ninths of what number? 68. Bought 2 barrels and 1 fifth of a barrel of gin (that is, 11 fifths of a barrel) for 65 dollars; what was that a barrel ? 69. 65 is 9 fifths of what number 1 70. 71 is 8 elevenths of what number” o ".. C. 1. A boy gave away 2 apples and 1 half, which was 1 fourth of all he had; how many had he 1 2. A man gave away 3 dollars, which was 2 fifths of all the money he had ; how much had he 3. A man sold a cow for fifteen dollars, which was 4 fifths of what she cost him ; how much did he lose by the bargain 4. A man sold a piece of cloth for 87 dollars, which was 9 eighths of what it cost him; how much did he gain by the bargain 5. There is a pole 3 fifths under water, and 7 feet out of the water; how long is the pole 1 6. A man sold a piece of cloth for 47 dollars, by which bargain he lost 2 ninths of what the cloth cost him; how much did it cost him, and how much did he lose 3 Miscellaneous Examples. 1. If a staff 5 feet long cast a shadow 4 feet at 12 o'clock, what is the length of a pole that casts a shadow 67 feet at the same time ! 2. If 53 gallons of water, in 1 hour, run into a cistern containing 97 gallons, and 44 gallons run out in an hour, in what time will it be filled ! 3. A man bought a cask of wine containing 75 gallons; 2 sevenths of it leaked out, and he sold the remainder for 1 dollar a gallon; how much did he sell it for't , 4. A cask of wine cost 67 dollars ; what is 5 eighths of it worth 1 5. A man bought 9 oranges for 6 cents and 2 sevenths apiece, and sold them for 67 cents; what did he gain by the bargain 6. A man bought 10 yards of broadcloth for 70 dollars; how must he sell it per yard in order to gain 14 dollars ? 7. If when the days are 12 hours long, a man perform a journey in 3 days, how many hours is he in performing it? 8. If a man perform a Journey in 36 hours, how many days would he be in performing it when the days are nine hours long? 9. If when the days are 11 hours long, a man can perform a journey in 5 days, in how many hours will he perform it? In how many days when the days are 9 hours long 1 10. What number added to 2 fifths of 33, will make the number 17 ? 11. How many yards of cloth, that is 1 quarter of a yard wide, will line 10 yards, that is 3 quarters wide : 12. 8 yards of cloth that is 1 quarter wide are equal to how many yards that is 4 quarters wide 1 13. How many yards of cloth, that is 3 quarters wide, are equal to 7 yards that is 5 quarters wide 3 14. How many yards of cloth, that is 6 quarters wide, are equal to 37 that is 4 quarters wide 1 15. If a piece of cloth, 5 quarters wide, be worth 37 dollars, what is a piece of the same length, 3 quarters wide, worth 16. If cloth 4 quarters wide is worth 8 dollars a yard, what is 1 yard of the same kind of cloth, that is 5 quarters wide, worth 1 SECTION XII. PARTs of one are called fractions. Fractions may be expressed by figures, as well as whole numbers. It requires two numbers to express a fraction; one, to show into how many parts one is divided; and the other to show how many of those parts are used. For example, if we wish to express one half, (which means that one is divided into two equal parts, and that one part is used) we must use the figure 2 to express that one is divided into two equal parts, and the figure 1 to show that one part is used. And these must be written in such a manner that we may always know what each of them is intended to express. One half is usually written thus ; ; one number above a line and the other below it. The number below the line shows into how many parts one is divided, and the number above the line shows how many parts are used. One third is written - - - # s Example. } of an apple signifies that the apple is to be cut into 7 equal parts, and that 3 parts are to be used. Let us apply an example to Plate II. # refers to a square divided into 8 parts, and signifies that 5 parts are to be used. We may observe, that when one is divided into 3 parts, the parts are called thirds, when one is divided into 4 parts, the parts are called fourths, &c. that is, the fraction takes its name from the number of parts into which one is divided. The number under the line is called the denominator, because it gives name to the fraction; and the number above the line is called the numerator, because it shows the number of parts used. Thus for, 10 is the denominator and 3 the numerator. N. B. The pupil must be made familiar with this mode of expressing fractions, and must be able to apply it to any familiar objects; as apples, oranges, &c.; and to the table, before he is allowed to proceed any farther. Particular care must be taken to make him understand what the denominator signifies, and what the numerator, as explained above. The denominator should always be explained first. The following examples are a recapitulation of some of the foregoing sections, for the purpose of showing the application of the above method of writing fractions. See Section VIII. A. A. 1. In 2 how many time 2. In 3 how many times ; 3. In 2 how many times 4. In 4 how many times 5. In 6 how many times 6. In 7 how many times 7. In 8 how many times 8. In 24f how many time 9. In 3; how many times #" 10. Reduce 4} to an improper fraction.} 11. Reduce 34 to an improper fraction. 12. Reduce 5; to an improper fraction. . * When the numerator is larger than the denominator, the fraction is called an improper fraction. #21-2 is read two and 1 half. It is called a mired number. i |