SECTION XIII. The operations in this section are the reducing of fractions to a common denominator, and the addition and subtraction of fractions. The examples will generally show what is to be done, and how it is to be done. Plate III will be found very useful m explaining the operatins, by exhibiting the divisions to the eye. 1. The first example may be illlustrated by the second square in the second row. This square is divided into halves by a vertical line, and then into fourths by the horizontal line. It will be readily seen that makes 2 fourths, and that the first had twice as much as the second. The plate will not be so' necessary for the practical questions as for the abstract. In the second example therefore it will be more useful than in the first. 4. It will readily be seen on the second square of the second row, that, and are 8. It will be seen in the third square of the sec that į makes %. 10 and 12. In the second square of the third row, it will be found, that makes ; and that ; make 4. 25. In the fourth square of the second row, it will be seen that I half is ; and in the second square of the fourth row, is , both together make A and į make z. 27. În the second square of the fourth row, i is the same as 33. In the fifth square of the fourth row, it will be seen that * (made by the vertical division) contains to ; and in the fourth square of the fifth row { contains , and contain ; and in the second square of the tenth row jó contains : When these questions are performed in the mind, Hae pupil will explain them as follows. He will ond row, 20 probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and of 20 is 8, and i'u of 20 is 2; therefore 1 is mo, is and jó 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 7 parts, the square, therefore, is divided into 56 parts; 3 of the vertical divisions, or contain žá: 51. 1 half is 1, and is j, which added together make. 61. is * 1 is 25, is zo, which added together make is 67. { is in, is it'a, which added together make 41; from 17 take , and there remains li, 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 íš, and is , and both together make for 1}. 86. ķis , and į is a. 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 con tain 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 } are $; making in the whole 8 bushels. 5. 6 and 2 are 8; and į and are 37 or 117. *hich 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. . 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 ai are equal to : 13. f . Find the squares which are divided into 56 parts; they are the Sth 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 į, as may be seen by looking on the 3d square of the 4th row; therefore Mi is equal to . SECTION UY. A. This section contains the division of fractions hý whole numbers, and the multiplication of one fraction by another. Though thiese 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 20 square is divided vertically into halves, and each of the halves is divided into lialves by the horizontal line ; į of is therefore of the whole. 3 and 4. In the third row, the 2d square shows that { of is . 16 and 17. In the 5th row, the 3d square shows that į of} is ii of the whole. 33. Since of a share signify 3 parts of a share, it is evident that į of the tiiree parts is 1 part, that 39). signify 9 pieces or parts, and it is evident that ļof 9 parts is 3 parts, that is . 43. We cannot take ţ of 5 pieces, therefore we must take ţ of ã, which is in, and is 5 times as much as ā, therefore { ofis . This may be readily seen on the plate. In the sixth row, third square, find á by the vertical division, then these being divided each into three parts by the horizontal division, and of each being taken, you will have a 52. In the 4th row, the 3d square shows that of is to, and must be twice as mucli, or ta. 54. in tire fifth row, the 3d square shows that į ofis , but inust be twice as much as ļ, there. foreof, are in 78. }. 79. 81 is 49,1 of 1 is as, consequently 4 of 4 is to, or 115. 36. We may say 1 of 8$ is 2, and 24 over, then %} is, and į of 2 is af, hence } of 89 is 234. 90. of 181 is 233, and is 3 times as much, or 731. B. 4. It would take 1 man 4 times of, or 375 days, and 7 men would do it in of that time, that is, in 51% days. SECTION XY. A. This section contains the divisions of whole uumbers 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 į 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 4. If of a bushel would last a week, of bushels would last 27 weeks ; but since it takes 1, it will last only fof 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 1 of that number, that is to 5 persons, and he would have 1 bushel left, which would be of enough for another. 23. 9f is , and 14 is y. If it had been only of a dollar a barrel, he night liave bought 66 barrels for 93 dollars, but since it was 4 a bar |