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 be has seen the use of the operation, to let him study it. It may be illustrated on Plate III in the following manner. 8. 11. 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 if of the whole square, because it is divided into fourths vertically ; therefore ii are equal to . 13. 4*. Find the squares which are divided into 56 parts; they are the 3th 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 jf is equal to 1. SECTION XIV. square shows 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 ; j of is therefore 1 of the whole. 3 and 4. In the third row, the 2d that of 4 is . 16 and 17. In the 5th row, the 3d square shows that į of į is i's of the whole. 33. Since of a share signify 3 parts of a share, it is evident that į of the three parts is 1 part, that 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 1 of 5 pieces, therefore wo must take ţ of j, which is t'a, and is 5 times as much as y, therefore { of is is. This may be readily seen on the plate. In the sixth row, third square, find by the vertical division, then theso being divided each into three parts by the hori. zontal division, and of each being taken, you will have to 52. In the 4th row, the 3d square shows that of + is j's, and į must be twice as much, or is. 56. In the fifth row, the 3d square shows that } of is 1'5, but must be twice as much as į, thereforeof, are is: 78. 87 is y, of y is . 79. 87 is 0, of 1 is ab, consequently 4 of is it, or 111 36. We may say { of 89 is 2, and 2 over, then 24 is, and of is 21, hence of 8 is 234. 90. 1 of 181 is 233, and 4 is 3 times as much, or B. 4. It would take 1 man 4 times 9, or 374 days, and 7 men would do it in of that time, that is in 51days. SECTION XV. A. This section contains the divisions of whole numbers by fractions, and fractions by fractions. 1. Since there are in 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 } 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. 61 is 27. If of a bushel would last a week, 6 bushels would last 27 weeks ; but since it takes 2, it will last only of the time, or 9 weeks. 13. If he had given j. 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 į of that number, that is to 5 persons, and he would have I bushel left, which would be of enough for another. 23. 9 is 86, and 14 is 4. If it had been only f of a dollar a barrel, he might have bought 66 barrels for 99 dollars, but since it was a bar rel, he could buy only th of that number, that is, 6 barrels. 25 and 26. Ans. 94. 31. 41 is 4, and 9 is y. Now is contained in y 48 times, and is contained only 4 part as many times, consequently only 2 or 24. B. 1. } is foi consequently 5 pounds can be bought for of a dollar. 3. is ', and is in. If he had given only is apiece, he could have given it to 9 persons, but since he gave the could give it to only 1 half as many, or 41 persons. 5. is it, and į is it. If a pound had cost s of a dollar, 14 pounds could be bought for fi of a dollar, but since it costs at, only { as many can be bought; that is, 43 pounds. 9. } is to, and lf is . If a bushel had cost it of a dollar, 65 bushels might have been bought, but since it cost is, only if part as much could be bought ; that is, 417 bushels. 12. is t, and I is it, it is contained in ii 15 times, but is contained only as many times ; that is, 3 times, Miscellaneous Example . 5. of a penny is of 4 farthings. Ans. 27 farthings. 6. á of 12 pence. Ans. 10 pence. ht 7. of 4 quarters is 2 quarters and of a quarter; of a quarter is of Ā nails, which is 1nails. Ans. 2 quarters, l} nails. 13. of 24 hours is 15 hours.' 14. of 24 hours is 14 hours and of an hour; 4 of 60 minutes is 24 minutes. Ans. 14 hours, 24 voinutes. 2 28. There being 4 farthings in a penny, 1 fær. thing is part of a penny. 30. 3 farthings is į of a penny. 31. I penny is la of a shilling, because there are 12 pence in a shilling. 34. 5 pence is of a shilling. 49. 2 farthings is to, or of a shilling. 5 farthings is of a shilling: 51. 1 penny is so of 1 pound. 7 pence is to of I£. 59. 39. 5d. is 41 pence, which is to of 1€. 75. 1 nail is jy of a yard, 5 nails is to of a yard. 89. 1 oz. is it of 1 lb. 15 oz, is to of 1 lb. 91. 1 lb. is z of 1 quarter. 9 lbs. is one 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 f. 122. One paid , the other . 123, 20 dollars were paid in the whole, one paid , another, and the third and the - 121, 3 and 4 and 5, are 12. The first put in its the second 4 the third je 129. 4 dollars for 2 months, is the same as 8 dol. lars, 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 en B 's, and Cat, 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 |