2. | of 18 are 24; 24 is | of 27; 4 of 35 is 5; 27 is 5 times 5 and f of & C. This article contains the multiplication table, in which the numbers from 10 to 20 are multiplied by the ten first numbers. SECTION VIII. Explanation of Plate II. Plate I, which has been used in the preceding sections, presents each unit as a simple object and undivided. Plate II, presents the units as divisible objects, the different fractions of which form parts, and sums of parts of unity. This plate is divided into ten rows of equal squares, and each row into ten squares. The first row is composed of ten empty squares, which are to be represented ■ > the pupil as entire units. The second row presents ten squares, each divided into two equal parts by a vertical line, each of these parts of course represents one half. In the third row, each square is divided into three equal parts, by two vertical lines, each part representing one third, Spc. to the tenth row, which is divided into ten equal parts, each part representing one tenth of unity. N. B. In plates II and III, the spaces and not the marks are to be counted. Be careful to make the pupil understand, 1st, that each square on the plate is to be considered as an entire unit, or whole one. 2d, explain the divisions into two, three, four, &c. parts. 3d, teach him to name the different parts. Make him observe that the name shows into how many parts one is divided and how many parts are taken, in the same manne as it does when applied to larger numbers. -f f°r example, shows that one thing is to be divided into 7 equal parts, and 4 of those parts are to be taken. 4th, make the pupil compare the different parts together, and observe which is* the largest. Ask him such questions as the following: Which are the smallest halves or thirds? Ans. Thirds. Why? Because, the more parts a thing is divided into, the smaller the parts must be. A. 15. On plate II., count two squares in the second row, and then ascertain the number of spaces or halves in them. There are 4 halves. 21. In the 2d row take 3 squares and 1 space in the 4th square; then count the spaces. Ans. 7 halves. 37. In the 3d row take 5 squares, and 2 spaces in the 6th; then count the spaces or thirds. Ans. 17 thirds. 54. In the 5th row take 6 squares, and 4 spaces in the 7th square; then count the spaces or fifths. Ans. 34 fifths. B. 2. This operation is the reverse of the last. In the 2d row count 4 spaces or halves, and see how many squares or whole ones it takes. It will take 2. 38. In the 9th row count 48 spaces or 9ths, and see how many squares or whole ones it takes. It will take 5 squares and 3 spaces in the 6th. Ans. 5 whole ones and f. SECTION IX. A. 2. \ signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times I third is 2 parts, or |. 6. 7 times \ is f, or 2£. 10. On the plate in the 3d row, 5 times § are V°» which takes 3 squares and 1 space. Ans. 3£. 24. In the 9th row take 4 spaces or 9ths, and repeat them 5 times, which will make \9, and will take 2 squares and 2 spaces. Ans. 2f. B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 2, which added to 8 make 10. 18. 4 times 3 are 12, and 4 times f are or three whole ones, which added to 12 make 15. 32. 2 times 3 are 6, and 2 times ^ are $, which added to 6 make 6-f. 40. 10 barrels of cider at 3 dollars and £ a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times £ is 5/, or 8 and f of a dollar. Ans. 38f dollars. C. 2. £ to each would be 3 times £, or f, which are 2j oranges. 3. V or 2 bushels. 4. 7 times J are y, or gallons. 5. 8 yards and f or 2 yards, that is, 10 yards. 6. 4 times 2 are 8, and 4 times f are !/, or 2|, which added to 8 make 10f bushels. 12. It would take 1 man 3 times as long as it would 3 men. Ans. 13| days. 14. 3 men would build 3 times as much as 1 man; and in 4 days they would build 4 times as much as in 1 day. Ans. 38f rods. 15. Ans. 12 yards. SECTION X. A. 21. £ of 1 is |. I of 2 is 2 times as much, or f. £ of 4 is or 1| i of 5 is |, or If. f of 6 is f, or 2. iof7isf,or2£. 27. i of 1 is ±. | of 2 is §. } of 3 is f. \ of 7 is |, or If. This manner of reasoning may be applied to any number. To find j of 38: it is y, for j of 38 is 38 times as much as | of 1, and \ of 1 is |, consequently | of 38 is \8, and 3T8 is 5f 40. To find | of a number, \ must be found first, and then § will be 2 times as much. £ of 7 is J, and 2 times f are y, or 4f. 74. £ of 50 is S9% or 5f ; % is 4 times as much; 4 times 5 are 20, 4 times f are V» or which added to 20 make 22§. Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Arts. If apiece. 3. J of 3 is |; | of a bushel apiece. 4. J of 7 is 4J; he gave away 4£ and kept 2f. 6. 1 half dollar a yard, or 50 cents. 7. i of 7 is |, or If; | of a dollar is f of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. £ of 8 is If. f of 100 is 33f. Ans. 1 dollar and 33f cents, or it is 1 dollar and 2 shillings. 9. iff 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and §, and 2 bushels will cost 5£ dollars, Ans. 5 dollars and 2 shillings, or 33| cents. 13. If 7 pounds cost 40 cents, 1 will cost 5f cents; 10 pounds will cost 57f cents. 16. 1 cock would empty it in 6 hours, and 7 cocks would empty it in } of 6 hours, or ^ of 1 hour, which is $ of 60 minutes; f oMBO minutes; is 51f minutes. SECTION XI. A. 2. 2 halves of a number make the number; consequently 1 and 1 half is the half of 2 times 1 and 1 half, which is 3. 15. 4^ is £ of 5 times 4 and 3, which is 22|. 17. 4$ is i of 9 times 4$, which is 39f B. 4. 5 is 3 times | of 5, which is f, or If. 30. If 8 is £ of some number, | of 8 is { of the same number. | of 8 is 2|, 2f is { of 4 times 2|, which is 10}; therefore 8 is £ of 10}. 40. If 8 is I of 8 is |; } of 8 is f, | is | of y, or Of; therefore 8 is ^ of 9 f. 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, 4 If. 69. i of 65 is 7f; 7f is \ of 5 times 7|, which 36|. 65isfof36£. C. 4. 37 is f of 32f, which taken from 37 leaves 4£. Ans. 4£ dollars. 5. 7 feet must be | of the whole pole. f 6. If he lost }, he must have sold it for £ of what it cost. 47 is $ of 60f Ans. 60 dollars and 42f cents. Miscellaneous Examples. £ 1. The shadow of the staff is f of the length of the staff; therefore the shadow of the pole is J of the length of the pole. 67 is £ of 83£. Ans. 83} feet. 2. 9 gallons remain in the cistern in 1 hour, will be filled in 10 hours and |; £ of 60 minutes |