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24. A boy having a quart of nuts, wished to di. vide them, so as to give one companion }, another t, and a third 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 left for himself?

25. is how many $? 1 is how many ? ; and and į are how many f?

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

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

28. A boy having a pound of almonds, said he intended giving of them to his sister, and I to his brother, and the rest to his mamma. His mamma smiling said she did not think he could divide them

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

29. j is how many ? 1 is how many ? ; and are how many ?

30. Mr. Goodman having a pound of raisins, said he would give Sarah }, and Mary 4, 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 ţ and 1 and f of them. How many twelfths would each have ?

31. { and and are how many ik?

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 it. 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. 1 is how many ? } is how many zt? is how many z ? to is how many zo?

34. f is how many to ?
35. 1 is how many ty?
36. į is how many f?
37. } how many I'?
38. are how many $?
39. & are how many mi?
40. s is how many It?
41. ž are how many 15

? 42. & are how many is? 43. are how many zt? 44. & are how many 3'5 ? 45. are how many 5t? 46. are how many 36? 47. are how many zo? 48. Reduce to sixths and to sixths. 49. Å and are how many ? 50. Reduce and to eiguchs. 51. and are how many f? 52. 1 and 1 are how many ? 53. and į are how many ? 54. 1 and are how many f? 55. j and į are how many f? 56. and are how many ? 57. 1 and 1 and } are how many f? 58. 1 and and I are how many it? 59. and are how many ? 60. į and į and į are how many tł? 61. and my and are how many z ? 62. 1 and 1 and fand * and are how many 4? 63. 1 and are how many ? 64. — and į are how many I'}? 65. į and are how many it? 66. less į are how many $?

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67. 3 and 4, less , are how many th? 68. Š less gare how many t's? 69. & less į are how many 24? 70. less are how many 3'5? 71. ž, and, and ], and is, less , are how many 72. }, and }, and , and it, and zo, less y, are how

many to? 73. 4 and are how many ? 74. g and are how many 36

? 75. and are how many o's?

When the denominators in two or more fractions are the same, the fractions are said to have a common denominator. Thus fand i 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 s and is 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 ț and and to a common denominator.

79. Reduce f and to a common denominator. 80. Reduce 4 and to a common denominator.

81. Reduce į and f and to a common denominator.

82. Add together and
83. Add together and K.
84. Add together and is
85. Add together į and fand
86. Subtract from 3.
87. Subtract from
88. Subtract & from .
89. Subtract from

B. 1. Mr. F. said he would give į 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 11 bushels of wheat to one man, 44 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 7 yards of one kind of cloth, and 68 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 4 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 is out of the water?

Ans. 4.

8. A man having undertaken to do a piece of work, did } of it the first day, } 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 } 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 ; a line divided into halves and then into sixths, will show that i is the same as 4, and as $; $,,, can therefore be reduced to 1, and a to . 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.* 2. Reduce ' to its lowest terms. 3. Reduce s 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 je to its lowest terms. 9. Reduce it to its lowest terms. 10. Reduce 8 to its lowest terms. 11. Reduce 35 to its lowest terms. 12. Reduce f' to its lowest terms. 13. Reduce 4% to its lowest terms. 14. Reduce } to its lowest terms.

Note. It will be seen by the above section that if both the numerator and denominator be multiplicd 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 omi it till after the next section.

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