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and then distributed them as George had desirod. How many twentieths did he give to each ?

33. į is how many zo? } is how many as? is how many to ? is is how many ?

34. is how many t
35. į is how many ?
36. į is how many ś?
37. į is how many is?
38. are how many ś?
39. are how many it?
40. į is how many tó!
41. are how many it's

? 42. ğ are how many na

? 43. 4 are how many i ? 44. $ are how many 35 45. are how many 36

? 46. į are how many 36 47. i are how many ty? 48. Reduce to sixths and if to sixths. 49. f and I are how many t! 50. Ředuce and to eighths. 51. į and į are how many ? 52. į and į are how many ? 53. and i are how many : ? 54. 1 and are how many ģ

? 55. į and ž are how many ; ? 56. į and į are how many 's? 57. and and I are how many ? 58. 1 and and jo are how many to

? 59. and į are how many ty? 60. į and į and are how many i? 61. and it and are how many to ? 62. and 1 and 1 and 1 and 1 are how many ? 63. į and are how many t ? 64. and į are how many 13 65. and į are how many t's

? 66. į less į are how many f?

67. and , less 1', are how many ti? 68. less are how many is? 69. less ž are how many ot? 70. less & are how many 3? 71. ), and , and }, and B, less g, are how many 72. t, and , and }, and to, and at, less }, are how many ?

73. 7 and f are how many ? 74. Á and are how many t? 75. 1 and ž are how many is?

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When the denominators in two or more fractions are the same, the fractions are said to have a common denominator. Thus and if have a common denominator. We have seen that, when two or more fractions have a common denominator, they may be added and subtracted as well as whole numbers. We add or subtract the numerators, and write their sum or difference over the common denominator. The first part of the process in the above examples was to reduce them to a common denominator.

76. Reduce and to a common denominator. Note. They may be reduced to twelfths.

If it cannot be immediately seen what number must be the common denominator, it may be found by multiplying all the denominators together ; for that will always produce a number divisible by all the denominators.

77. Reduces and to a common denominator.

78. Reduce ž and 1 and $ to a common denominator.

79. Reduce } and to a common denominator. 80. Reduce j and to a common denominator.

81. Reduce and f and if to a commou denomiDator.

82. Add together and .
83. Add together and it
84. Add together f and pas
85. Add together and fand po
86. Subtract from f.
87. Subtract is from g.
88. Subtract from .
89. Subtract s from $.

B. 1. Mr. F. said he would give of a pine-apple to Fanny, and ? to George, and the rest to the one that could tell how to divide it, and how much there would be left. But neither of them could tell; so he kept it himself. Could you have told if you had been there? How would you divide it? How much would be left ?

2. A man sold 17 bushels of wheat to one man, 4 bushels to another; how many bushels did he sell to both ?

3. A man bought 64 bushels of wheat at one time, and 2} at another; how much did he buy in the whole ?

4. A man bought 74 yards of one kind of cloth, and 64 yards of another kind; how many yards in the whole

5. A man bought of a barrel of flour at one time, 2} barrels at another, and 64 at another; how much did he buy in the whole ?

6. A man bought one sheep for 49 dollars, and another for 50 dollars; how much did he give for both ?

7. There is a pole standing, so that of it is in the mud, and f of it in the water, and the rest out of the water ; how much of it is out of the water ?

8. A man having undertaken to do a piece of work, did of it the first day, į of it the second day, and ; of it the third day, how much of it did he do in three days ?

9. A man having a piece of work to do, hired two men and a boy to' do it. The first man could do į of the work in a day, and the other of it, and the boy } of it; how much of it would they all do in a day?

Note. By dividing a line into halves, and then into fourths, it will be seen that is the same as 1, a line divided into halves and then into sixths, will show that I is the same as }, and as 3; 4, 5, can therefore be reduced to }, and to go This is called reducing fractions to their lowest terms. It is done by dividing the greatest number that will divide it without a remainder. 1. Reduce to its lowest terms.*

Ans. . 2. Reduce so to its lowest terms. 3. Reduce & to its lowest terms. 4. Reduce to its lowest terms. 5. Reduce ii to its lowest terms. 6. Reduce is to its lowest terms. 7. Reduce to its lowest terms. 8. Reduce to its lowest terms. 9. Reduce if to its lowest terms. 10. Reduced to its lowest terms. 11. Reduce it to its lowest terms. 12. Reduce f} to its lowest terms. 13. Reduce to its lowest terms. 14. Reduce 41 to its lowest terms.

Note. It will be seen by the above section that if both the numerator and denominator be multi plied by the same number, the value of the fraction will not be altered; or if they can both be divided ly the same number without a remainder, the frạction will not be altered.

* If this article should be found too difficult for the pupil, he may omit it til after the next section.

SECTION XIV.

A. I A Boy having of an orange gave away of that, what part of the whole orange did he give

away ?

2. What is ļof į?

3. If you cut an apple into three pieces, and then cut each of those pieces into two pieces, how many pieces will the whole apple be cut into ? What part of the whole apple will one of the pieces be ?

4. What is off?

5. A boy had of a pine apple, and cut that half into three pieces, in order to give away { of it. What part of the whole apple did he give away ?

6. What is į of į?

7. If an orange be cut into 4 parts, and then each of the parts be cut in two, how many pieces will the whole be cut into !

8. What is of į?

9. A man having į a barrel of flour, sold of that; how much did he sell ?

10. What is of į?

11. If an orange be cut into 4 equal parts, and each of those parts be cut into 3 equal parts, how many parts will the whole orange be cut into ?

12. What is į of į?

13. A boy having į of a quart of chestnuts, gave away of what he had. What part of the whole quart did he give away ?

14. What is of į? 15. What is į of į? 16. A man owning of a ship, sold of his share ; what part of the ship did he sell, and what part did he then own ?

17. What is į of } ? 18. What is į of ?

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