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B. 4. 12 is of what number? That is, 12 is 3 sevenths of what number ? Ars. 28.

12. 4 is of what number? That is, 4 is 3 fifths of what number ? Ans. 64.

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Explanation of Plate III.

Plate III is intended to represent fractions of unity, divided into other fractions ; it is, therefore, an extension of plate II. It differs from it, only in this, that besides the vertical divisions, the squares are divided horizontally, so as to cut the fractions of the square into fractions of fractions. The horizontal lines are dotted, but they are to be considered as lines.

This plate, like the preceding, is divided into ten rows of squares, each row containing ten equal squares. In the first row, the first square is undivided, the 9 following squares are divided by hori-zontal lines into from two to ten equal parts. In all the other squares

the vertical divisions are the same as in Plate II, and besides this, each row is divided - horizontally in the same manner as the first row.

By means of this double division, the 2d row presents a series of fractions, from halves to twentieths. The 3d row presents a series from thirds to thirtieths, and so on to the 10th row, which presents a series from tenths to hundredths.

The 21 row, besides presenting halves, fourths, sixths, eighths, &c. shows also halves of halves, thirds of halves, fourths of halves, &c. and shows their ratios with unity.

The 3d row, bosides thirds, sixths, ninths, &c. shows halves of thirds, thirds of thirds, &c. and their ratios with unity. The other rows present analagous divisions

SECTION XIII.

The operations in this section are the reducing of fractions to a common denominator, and the ad. dition ahd 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 in explaining the operations, hy exhibiting the divisions to the eye.

1. The first example may be illustrated 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 .

8. It will be seen in the third square of the second row, that į makes à.

10 and 12. In the second square of the third row, it will be found, that { makes; and that make t.

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, į is , both together make and į make .

27. In the second square of the fourth row, i is the sanje as .

33. In the fifth square of the ruurth row, it will be seen that ; (made by the vertical division) contains zr; and in the fourth square of the fifth row

contains , and contain ; and in the second square of the tenth row is contains no

When these questions are performed in the mind, the pupil will explain them as follows. He will

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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 y of 20 is 2; therefore is one is

and to is to. 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 it.

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

61. fis i' is is , which added together make

67. is, is , which added together make 17; from 17 take it, 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 is, and both together make 1 or 14.

86. jis, and } is 3. If 4 be taken from there remains š.

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 1,9 the whole numbers. 6 and 2 are 8; 1 half and I are , making in the whole 85 bushels.

5. 6 and 2 are 8; and į and are j} or 117, which joined with 8 make 911.

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 fo!!owing manner.

8. ja. 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 .

13. ff. 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 f may be still reduced to }, as may be seen by looking on the 3d square of the 4th

row;

therefore It 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 } is therefore of the whole.

3 and 4. În the third row, the 2d square shows that į ofis :

16 and 17. In the 5th row, the 3d square shows that } of } is i' 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

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

43. We cannot take 1 of 5 pieces, therefore we must take ţ of t, which is t'a, and \ is 5 times as much as d, therefore { of { is . 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 į of each being taken, you will

52. In the 4th row, the 3d square shows that } of + is 1s, and must be twice as much, or in

56. In the fifth row, the 3d square shows that of f is is, but must be twice as much as ļ, there fore of 3, are in

have it.

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