1 2. of 18 are 24; 24 is of 27; 4 of 35 is 5; 27 is 5 times 5 and of 5. C. This article contains the multiplication table, in which the numbers from 10 to 20 are inultiplied by the ten first numbers. SECTION YIII. Expionation 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 to 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, fc. 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 divi. sions 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. for 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 6t3. Ans. 5 whole ones and . SECTION IX. A. 2. signifies that I thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times I third is 2 parts, or . 6, 7 times į is j, or 2. 10. On the plate in the 3d row,5 times are , 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 ", and will take 2 squares and 2 spaces. Ans. 24. 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 are 4, 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 64. 40. 10 barrels of cider at 3 dollars and a bar 10 barrels at 3 dollars, would be 30 dollars, then 10 times į is , or 8 and of a dollar. Ans. 38f dollars. C. 2. to each would be 3 times , or, which are 21 oranges. 3. or bushels. 6. 4 times 2 are 8, and 4 times are 1, or 23, which added to 8 make 10% 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 I man; and in 4 days they would build 4 times as much as iu 1 day. Ans. 38, rods. 15. Ans. 12 yards. rel; SECTION X. A. 21. j of 1 is į of 2 is 2 times as much, or . į of 4 is , or 1} } of 5 is , or 1}. of 6 is , or 2. of 7 is j, or 2. 27. } of 1 is š. } of 2 is g. } of 3 is of 7 is j, or lz This manner of reasoning may be applied to any number. To find į of 38: it is.38, for of 38 is 38 times as much as of 1, and of 1 is –, consequently 4 of 38 is , and is 51. 40. To find į of a number, į must be found first, and then will be 2 times as much, of 7 is }, and 2 times are «, or 43. 74. ) of 50 is 50, or 5; is 4 times as much; A times 5 are 20, 4 times are ", or 2%, 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. Ans. If apiece. 7. § of 7 is j, or 14 ; of a dollar is of 100 cents, which is 40 cents. Ans. I dollar and 40 cents a bushel. 8. ở of 8 is 1. of 100 is 33. Ans. 1 dollar and 33 cents, or it is 1 dollar and 2 shillings. 9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and, and 2 bushels will cost 51 dollars. Ans. 5 dollars and 2 shillings, or 33; cents. 13. If 7 pounds cost 40 cents, I will cost 54 cents; 10 pounds will cost 574 cents. 16. 1 cock would empty it in 6 hours, and 7 cocks would empty it in 4 of 6 hours, or of 1 hour, which is of 60 minutes ; 4 of 60 minutes; is 517 minutes SECTION XI. A. 2. 2 halves of a number make the number; consequently 1 and 1 half is the half of 2 times i and 1 half, which is 3. 15. 4. is į of 5 times 4 and , which is 224. 30. If 8 is of some number, į of $ is of the same number of 8 is 2, 2 is 1 of 4 times 2, which is 103; therefore 8 is of 10%. 40. If 8 is 4, 5 of 8 is ; of 8 ist, is 4 of *, or 93; therefore 8 is of 9 %. 52. If of a ton cost 23 dollars, j of a ton must be'} of 23, that is 4 dollars, and the whole would cost 9 times as much, that is, 414. 69. } 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 41. Ans. 4f dollars. 5. 7 feet must be of the whole pole. 6. If he lost , he must have sold it for J of what it cost. 47 is 7 of 60. Ans. 60 dollars and 429 cents. Miscellaneous Elampies. 1. The shadow of the staff is of the length of the staff; therefore the shadow of the pole is of the length of the pole. 67 is of 834. Ans. 834 feet. 2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7; of 60 minutes |