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 I are a. 8. It will be seen in the third square of the second row, that ļ makes . 10 and 12. In the second square of the third row, it will be found, that ļ makes; and that make 4. 25. In the fourth square of the second row, it will be seen that 1 half is ; and in the second square of the fourth row, I is , both together make and make 1. 27. În the second square of the fourth row, is the same as 33. In the fifth square of the fourth row, it will be seen that ? (made by the vertical division contains ; and in the fourth square of the fifth row contains , and contain ; and in the second square of the tenth row contains When these questions are performed in the mind, He pupil will explain them as follows. He will probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and of 20 is 8, and 7 of 20 is 2; therefore is , is and i'o 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 .. 51. 1 half is ,, and 1 is , which added together make . 61. is amor yo is is no, which added together make 67. { is iisia, which added together make 11; from 17 take y, and there remains 19, 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 iš, and is , and both together make us or 1.. 86. jis ž, and į is . If 4 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 85 bushels. 5. 6 and 2 are 8; j and į and I are i7 or 147 which joined with 8 make 9ji. 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 IS 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 1 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 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 i, as may be seen by looking on the 3d square of the 4th row; therefore -** is equal to 1. SESTION UY. A. This section contains the division of fractious hý 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 20 square is divided vertically into halves, and each of the halves is divided into lialves by the horizontal line ; 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 of is of the whole. 33. Since of a share signify 3 parts of a share, it is evident that of the tiiree parts is 1 part, that is 1. 39. o signify 9 pieces or parts, and it is evident that ļof 9 parts is 3 parts, that is . 43. We cannot take 1 of 5 pieces, therefore we must take of 1, which is i'a, and is 5 times as much as ā, therefore of 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 take, you will "have 52. In the 4th row, the 3d square shorvs that of i is , and must be twice as much, or . 50. in the fifth row, tbe 3d square shows that of is , but sinust be twice as much as ļ, there. fore of, are in 78. 9 is y, i of vis . 79. 81 is 494 of 4 is aó, consequently of a is 4, or 11. 36. We may say 1 of 8f is 2, and 24 over, then 2 is 2, and į of 2 is 4, hence ; of 89 is 234. 90. of 183 is 23, and is 3 times as much, or 71. B. 4. It would take 1 man 4 times 93, or 375 days, and 7 men would do it in of that time, that is, in 5 days. SECTION XY. 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. 17. 61 is . If of a bushel would last a week, 6 bushels would last 27 weeks ; but since it takes *, 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. 99 is 00, and 14 is V. If it had been only of a dollar a barrel, he night have bought 66 barrels for 9 dollars, but since it was ^ a bar |