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probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and f of 20 is 8, and to of 20 is 2; therefore { is mois

and to is . All the examples should be explained in the same manner.

45. In the 8th row, the 7th square is divided vertically into 8 parts, and horizontally into a parts, the square, therefore, is divided into 56 parts; 3 of the vertical divisions, or contain t.

51. 1 half is, and is , which added together

makes

.

61. f is also jy is also f is so, which added togetirer make 16

67. is isis, which added together make H; from 17 take is, and there remains it, or 1.

82. It will be easily perceived that these examples do not differ from those in the first part of the section, except in the language used. They must be reduced to a common denominator, and then they may be added and subtracted as easily as whole numbers. is Is, and is t, and both together make if or 14.

86. is, and is . If be taken from there remains

B. This article contains only a practical application of the preceding.

3. This example and some of the following contain mixed numbers, but they are quite as easy as the others. The whole numbers may be added separately, and the fractions reduced to a common denominator, and then added as in other cases, and afterwards joined to the whole numbers. 6 and 2 are 8; 1 half and I are , making in the whole 83 bushels.

5. 6 and 2 are 8; and į and are j} or 117, which joined with 8 make 917.

C. It is difficult to find examples which will aptly illustrate this operation. It can be done more conveniently by the instructer. Whenever a fraction occurs,

which may be reduced to lower terms, if it be suggested to the pupil, he will readily perceive it and do it. This may be done in almost any part of the book, but more especially after studying the 13th section. Perhaps it would be as well to omit this article the first time the pupil goes through the book, and after he has seen the use of the operation, to let him study it. It may be illustrated on Plate III in the following manner.

8. j. Find all the squares which are divided into 24 parts. There are 4 squares which are divided into 24 parts, viz. the 8th in the 3d row, the 3d in the 8th row, the 6th in the 4th row, and the 4th in the 6th row. Then see if exactly 18 can be found in one or more of the vertical divisions. In the 6th square of the 4th row, there are exactly 18 divisions in three vertical divisions, but those 3 vertical divisions are of the whole square, because it is divided into fourths vertically; therefore if are equal to .

13. 3. Find the squares which are divided into 56 parts; they are the 8th in the seventh row, and the 7th in the 8th row ; see if in either of them, one or more of the vertical divisions contain exactly 42 parts. In the 7th of the 8th row, 6 vertical divisions contain exactly 42; these divisions are of the square, for it is divided vertically into 8 parts. But may be still reduced to }, as may be seen by looking on the 3d square of the 4th row; therefore 13 is equal to 1.

SECTION XIV.

A. This section contains the division of fractions by whole numbers, and the multiplication of one fraction by another. Though these operations sometimes appear to be division, and sometimes multiplication, yet there is actually no difference in the operations.

The practical examples will generally show how the operations are to be performed, but it will be well to use the plate for young pupils.

1 and 2. In the second row, the 2d square is divided vertically into halves, and each of the halves is divided into halves by the horizontal line ; į of į is therefore 1 of the whole.

3 and 4. In the third row, the 2d square shows that ofis

을 16 and 17. In the 5th row, the 3d square shows . that ļof } is 's of the whole.

33. Since of a share signify 3 parts of a share, it is evident that of the three parts is 1 part, that j

1

is

.

39. signify 9 pieces or parts, and it is evident that } of 9 parts is 3 parts, that is f.

43. We cannot take of 5. pieces, therefore we must take of , which is is, and is 5 times as much as , therefore { of gis . This may be readily seen on the plate. In the sixth row, third square, find i by tho vertical division, then these being divided each into three parts by the horizontal division, and of each being taken, you will

have it

52. In the 4th row, the 3d square shows that of + is tg, and į must be twice as much, or a.

56. In the fifth row, the 3d square shows that of f is t's, but must be twice as much as ļ, there fore & off, are in

78. 87 is y, sofy is .

79. 87 is 40, 1 of 4 is a, consequently 4 of is , or 111.

36. We may say of 8f is 2, and 2. over, then 24 is , and į ofis 34, hence of 8% is 234.

90. of 18} is 23}, and is 3 tiines as much, or 731

B. 4. It would take 1 man 4 times 9, or 375 days, and 7 men would do it in ¢ of that time, that is, in 5** days.

SECTION XV.

a

а

A. This section contains the divisions of whole numbers by fractions, and fractions by fractions.

1. Since there are f in 2, it is evident that he : could give them to 6 boys if he gave them i apiece, but if he gave them i apiece, he could give them to only one half as many, or 3 boys.

5. If of a barrel would last them one month, it is evident that 4 barrels would last 20 months, but since it takes of a barrel, it will last them but one half as long, or 19 months.

7. 64 ie . If of a bushel would last a week, 64 bushels would last 27 weeks ; but since it takes , it will last only of the time, or 9 weeks.

13. If he had given į of a bushel apiece, he might have given it to 17 persons, but since he gave 3 Lalves apiece, he could give it to only of that: number, that is to 5 persons, and he would have I bushel left, which would be of enough for another.

용 23. 9is 6, and 14 is y. If it had been only 4 of a dollar a barrel, he might have bought 66. barrels for 97 dollars, but since it was y a bar

.

?

rel, he could buy only t of that number, that is, 6 barrels.

25 and 26. Ans. 94.

31. 4} is , and 93 is 48. Now } is contained in 48 48 times, and is contained only 1 part as many times, consequently only 20 or 24.

B. 1. į is si consequently 5 pounds can be bought for of a dollar.

1 / 3. is is, and is . If he had given only in apiece, he could have given it to 9 persons, but since he gave if he could give it to only 1 half as many, or 41 persons.

5. is, and is it. If a pound had cost of a dollar, 14 pounds could be bought for it of a dollar, but since it costs, only į as many can be bought; that is, 4 pounds.

9. is 16, and it is . If a bushel had cost ad of a dollar, 65 bushels might have been bought, but since it cost 16, only is part as much could be bought ; that is, 4 bushels.

12. f is na, and ž is in this contained in if 15 times, but iis contained only { as many times ; that is, 37 times.

Miscellaneous Examples.

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5. f of a penny is of 4 farthings. Ans. 24 farthings.

6. of 12 pence. Ans. 10 pence.

7. of 4 quarters is 2 quarters and of a quarter; } of a quarter is of 4 nails, which is if nails, Ans. 2 quarters, 1 nails.

13. 1 of 24 hours is 15 hours. 14. of 24 hours is 14 hours and of an hour ; of 60 minutes is 24 minutes. Ans. 14 hours, 24 minutes.

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