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24. A boy having a quart of nuts, wished to divide them, so as to give one companion ļ, another 1, and a third į of them ; but in order to make a proper division, he first divided the whole into eight equal parts, and then he was able to divide them as he wished. How many eighths did he give to each ? How many eighths had he ieft for himself?

25. Jis how many } ? is how many 1? and and į are how many į?

26. A man gave of a barrel of flour to one man, and of a barrel to another; to which did he give the most ? How much ?

27. Which is the largest or ? How much the largest ?

28. A boy having a pound of almonds, said he intended to give of them to his sister, and to his brother, and the rest to his mamma.

His mamma smiling said she did not think he could divide them so.

O yes I can said he, I will first divide them into twelve equal parts, and then I can diviile them well enough. Pray how many twelfths did he give to each ?

29. { is how many in? | is how many 'a ? } and į are how many iz ?

30. Mr. Goodman having a pound of raisins, said he would give Sarah ḥ, and Mary 1, and James } of them, and he told Charles he should have the rest, if he could tell how to divide them. Well, said Charles, I would first divide the whole into twelve equal parts, and then I could take , and į and of them. How many twelfths would each have ?

31. j and and are how many i's ?

32. George bought a pine apple, and said he would give of it to his papa, and ; to his mamma, and to his brother James, if he could divide ii. James took it, and cut it into twenty equal pieces,

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and then distributed them as George had desired. How

many twentieths did he give to each ? 33. I is how many go? į is how many to? f is how many z'o ? is is how many z' ?

34. is how niany to
35. 1 is how many i
36. į is how many $?
37. į is how many is?
38. į are how many $?
39. are how many a
40. Į is how many i'o.
41. are how muny is

?
42. ă are how many.i's
43. 4 are how many
44. are how many 's ?
45. are how many ?
46. į are how many 3'.
47. inre how many zo

? 48. Reduce į to sixths and į to sixths. 49. i and jare how many ,

? 50. Reduce and to eighths. 51. and are how many ś? 52. į and are how

? 53. į and į are how many ! 54. and į are how many

? 55. and I are how many į? 56. į and are how many is? 57. and 1 and 1 are how many } ! 58. and and iare how many is? 59. į and are how many 'a ? 60. į and 1 and 1 are how many is? 61. and in and are how many zo ? 62. and and and 1 and i' are how many is? 63. and are how many it? 04. and į are how many it's

? 65. į and } are how many 1'3 66. less are how many ;

many i

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67. and less are how manyta ?
68. less are how many is?
69. less are how many of

? 70. f less are how many 71. į and I, and y, and , less are how many 72. & and j, and, and it, and id, less are how many to ?

73. 4 and i are how many is! 74. and are how many

? 75. and are how many o'z

? When the denominators in two or more fractions are the same, the fractions are said to have a common denominator. Thus 4 and 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 f 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. Reduce and to a common denominator.

78. Reduce and i and to a common denoininator.

79. Reduce } and to a common denominator. 80. Reduce 1 and į to a common denominator

81. Reduce 1 and j and to a common denominator.

82. Add together and f.
83. Add together and it
84. Add together and
85. Add together į and 1 and 1's.
86. Subtract į from ź.
87. Subtract is from
88. Subtract from .
89. Subtract from f.

B. 1. Mr. F. said he would give { of a pine ap ple 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 1} bushels of wheat to one man, 4 bushels to another ; how many bushels did he sell to both ?

3. A man bought 6 bushels of wheat at one time, and 2į at another. How much did he buy in the whole ?

4. A man bought 78 yards of one kind of cloth, and 6 yards of another kind ; how many yards in the whole ?

5. A man bought of a barrel of beer at one time, 27 barrels at another, and 6; at another; how much did he buy in the whole ?

6. A man bought one sheep for 44. dollars, and another for 54 dollars; how much did he give for both ?

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

8. A man having undertaken to do a piece of work, did } of it the first day, i of it the second day, and I 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?

C. It will be seen by looking on plate III, that is the same as ļ, and that is the same as ļ, and that f is the same as ș ; 1, , can therefore be reduced to y, and to. This is called reducing fractions to their lowest terms. 1. Reduce š to its lowest terms.*

Ans. 2. Reduce o to its lowest terms. 3. Reduce to its lowest terms. 4. Reduce to its lowest terms. 5. Reduce in to its lowest terms. 6. Reduce to its lowest terms. 7. Reduce to its lowest terms. 8. Reduce to its lowest terms. 9. Reduce to its lowest terms. 10. Reduce to its lowest terms. 11. Reduce jk to its lowest terms. 12. Reduce i to its lowest terms. 13. Reduce to its lowest terms. 14. Reduce 43 to its lowest terms.

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

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

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