would empty it in 4 of 6 hours, or $ of 1 hour, which is of 60 minutes ; $ of 60 minutes; is 514 minutes. SECTION XI. A. 2. 2 halves of a nuniber make the number ; consequently 1 and 1 half is the half of 2 times i and I half, which is 3. 15. 41 is į of 5 times 4 and ļ, which is 224. 30. If 8 is of some number, j of 8 is of the same number. of 8 is 25, 2; is 1 of 4 times 2, which is 103 ; therefore 8 is of 104. 40. If 8 is 1, 3 of 8 is † ; š of 8 is , : is 4 of 4, or 9%; therefore 8 is of 9. 52. If of a ton cost 23 dollars, of a ton must be } of 23, that is 4 dollars, and the whole would cost 9 times as much, that is, 413. 19. \ of 65 is 7; 7 is į of 5 times 7%, which is 36. 65 is of 36. C. 4. 37 is of 32, which taken from 37 leaves 4. Ans. 4; dollars. 5. 7 feet must be of the whole pole. (i. If he lost ý, he must have sold it for of what it cost. 47 is 7 of 604. Ans. 60 dollars and 424 cents. Miscellaneous Eramples. 1. The shadow of the staff is of the length of the staff; therefore the shadow of the pole is fof the length of the pole. 67 is of 83. Ans. 831 feet. 2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7; 7 of 60 minutes are 46 minutes and %; of 60 seconds are 40 ser :onds. Ans. 10 hours, 46 minutes, 40 seconds. 10. Find of 33, and subtract it from 17. Ans. 3:. 11. It will take 3 times 10 yards. 13. 5 is of 3; it will take as much. Or 7 yards, 5 quarters wide, are equal to 35 yards 1 quai ter wide, which is equal to 11; yards that is :) quarters wide. 15. of 37 dollars. 16. as much. SECTION XII. The examples in this section are performed in: precisely the same manner, as those in the sections: to which they refer. All the difficulty consists in comprehending, that fractions expressed in figures': signify the same thing as when expressed in words.. Make the pupil express them in words, and all the difficulty will vanish. Let particular attention be paid to the explanation of fractions given in the section. VIII. A. 6. In 7 how many f? expressed in words, is, in 7 how many sixths ? Ans. 2. 14. Reduce 8 to an improper fraction; that is, in 8 and 3 tenths, how many tenths ? Ans. jā. B. 8. 3 are how many times 1 ? That is, in 23: sevenths how many whole ones ? Ans. 34. IX. B. 3. How much is 5 times 61 ? That is,, how much is 5 times 6 and 4 sevenths ? Ans. 32. V. & X. 15. What is of 27 ? That is, what is: 5 eighths of 27 ? Ans. 167. VI. & XI. A. 8. 74 is of what number ? That: is, 7 and 6. sevenths is 1 eighth of what number 1 Ans. 624 B. 4. 12 is of what number 1 That is, 12 is 3 sevenths of what number ? Ans. 28. 12. 4 is of what number? That is, 4 is 3 fifths of what number ? Ans. 6. 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 horizontal 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 2d_row, besides presenting halves, fourths, sixths, eightlis, &c. shows also halves of halves, thirds of halves, fourths of lialves, &c. and shows their ratios with unity. The 3d row, besides 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 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 in explaining the operations, by exhibiting the divisions to the eye. 1. The first example may be illustrated by the second square in the second row. 'I his square is divided into lialves by a vertical line, and then into fourths by the horizontal line. It will be readily seen that makes 2 fourthis, 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 inore useful than in the first. 4. It will readily be seen on the second square of the second row, that and d are . 8. It will be seen in the third square of the second row, that į makes a . 10 and 12. In the second square of the third row, it will be found, that į makes ; and that make . 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, is , both together make es and make 7. 27. In the second square of the fourth row, is the same as a 33. In the fifth square of the fourth row, it will be seen that + (made by the vertical division) contains at; and in the fourth square of the fifth row { contains go, and contain an; and in the second square of the tenth row i'ó contains When these questions are performed in the mind, the pupil will explain them as follows. He will 9 probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and of 20 is 8, and is of 20 is 2; therefore 1 is an, is and so is go: 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, and is , which added together make 61. is zo, iis , is to, which added together make 67. ķis is t, which added together make 17; from ;} take in, and there remains là, or 1. 82. It will be easily perceiyed 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 i's, and both together make is or 1}. 86. is, and is *. If y be taken from there remains & B. This article contains only a practical applica; tion 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 to the whole numbers. 6 and 2 are 8; 1 half and į are 5, making in the whole 81 bushels. 5. 6 and 2 are 8; ; and į and I are j7 or 117, which joined with 8 make 937. |