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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. H. 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 are equal to 1.

13. 42. 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 f 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 H is equal to

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; 1 of 1 is therefore 1 of the whole.

3 and 4. In the third row, the 2d square shows that 1 of 4 is.

16 and 17. În the 5th row, the 3d square shows that of is I'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 is, 1

39. , signify 9 pieces or parts, and it is evident that 1 of 9 parts is 3 parts, that is, .

43. We cannot take 1 of 5 pieces, therefore we must take f of t, which is it, and is 5 times as much as t, therefore of g is is. This may be readily seen on the plate. In the sixth row, third square, find by the vertical division, then these being divided each into three parts by the horizontal division, and of each being taken, you will

have ਭ•

52. In the 4th

row,
the 3d
square

shows that f of is th, and must be twice as much, or 56. In the fifth row, the 3d square shows that of is , but must be twice as much as $, therefore i of } are 15

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78. 87 is , & of 4 is 3.

79. 87 is, of is to, consequently 7 of is f$, or 14

86. We may say $ of 89 is 2, and 2 over; then 24 is , and 1 of 2 is 3%; hence of 89 is 234.

90. 4 of 18is 23, and 4 is 3 times as much, or

B. 4. It would take 1 man 4 times 94, or 37 days, and 7 men would do it in 7 of that time, that is, in 5 days.

SECTION XV.

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

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

5. If t 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 10 months.

7. 64 is y. If of a bushel would last a week, 64 bushels would last 27 weeks; but since it takes 1, it will last only of the time, or 9 weeks.

13. If he had given £ of a bushel apiece, he might have given it to 17 persons; but since he gave 3 halves apiece, he could give it to only of that number, that is, to 5 persons, and he would have 1 bushel left, which would be of enough for another.

23. 94 is 66, and 14 is y. If it had been only

of a dollar a barrel, he might have bought 66 barrels for 93 dollars · but since it was y a bar

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

25 and 26. Ans. 94.

31. 4£ is ', and 9f is . Now is contained in ¥ 48 times, and is contained only at part as many times, consequently only 2 or 24.

B. 1. is so i consequently 5 pounds can be bought for of a dollar.

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

5. is at, and f is 14. If a pound had cost a of a dollar, 14 pounds could be bought for 14 of a dollar; but since it costs 1, only as many can be bought; that is, 47 pounds.

9. is 18, and 13 is ft. If a bushel had cost ut of a dollar, 65 bushels might have been bought; .but since it cost 18, only to part as much could be bonght; that is, 474 bushels.

12. g is ts, and is t ; I's is contained in tf 15 times, but is is contained only t as many times ; that is, 37 times.

Miscellaneous Examples.

5. if of a penny is $ of 4 farthings. Ans. 24 farthings.

6. k of 12 pence. Ans. 10 pence.

7. of 4 quarters is 2 quarters and f of a quarter, of a quarter is fof nails, which is 14 naile. Ans. 2 quarters, 1f nails.

13. f of 24 hours is 15 hours. 14. šof 24 hours is 14 hours and f of an hour; of 60 minutes is 24 minutes. Ans. 14 hours, 4 minutes.

28. There being 4 farthings in a penny, 1 farthing ist part of a penny.

30. 3 farthings is of a penny.

31. 1 penny is I of a shilling, because there are 12 pence in a shilling.

34. 5 pence is of a shilling.
41. 1 shilling is at of a pound.
43. 3 shillings is of a pound.
48. i farthing is of one shilling.

49. 2 farthings is to, or 4 of a shilling. 5 farthings is of a shilling.

51. 1 penny is ado of 1 pound. 7 pence is zło of 1€.

59. 3s. 5d. is 41 pence, which is of 1£. 75. 1 nail is to of a yard. 5 nails is of a yard. 89. 1 oz. is it of 1 lb. 15 oz. is 1 of 1 lb.

91. 1 lb. is zo of 1 quarter. 9 lbs. is as of 1 quarter.

100. At the end of 1 hour they would be 7 and miles apart. In 7 hours, 7 times 74, which is 54 miles.

121. This is the principle of fellowship; 3 shillings were paid ; one paid }, the other g.

122. One paid f, the other f.

123. 20 dollars were paid in the whole; one paid , another way, and the third

121, 3 and 4 and 5 are 12. The first put in ; the second ta ; the third B.

129. 4 dollars for 2 months is the same as 8 dollars for 1 month; 3 dollars for 3 months is the same as 9 dollars for 1 month; and 2 dollars for 4 months is the same as 8 dollars for 1 month. The question is the same as if A had put in 8 dollars, B. 9 dollars, and C. 8 dollars. A must have As, B

, and C, of 100 dollars. 131. A's money was in 4 times as long as C's. It is the same as if A had put in 8 dollars for the

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