2. 4 of 18 are 24; 24 is † of 27; 4 of 35 is 5; 27 is 5 times 5 and 3 of 5. 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, &c. 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- 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 3. A SECTION IX. A. 2. signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or 3. 6. 7 times is 1, or 21. 10. On the plate in the 3d row, 5 times are 1, 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 2o, and will take 2 squares and 2 spaces. Ans. 23. 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 12, 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 69. 40. 10 barrels of cider at 3 dollars and & a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times § is 5o, or 8 and of a dollar. Ans. 382 dollars. C. 2. 2 to each would be 3 times 2, or, which are 24 oranges. 3.1 or 2 bushels. 4. 7 times are 2, or 51 gallons. 5. 8 yards and or 2 yards, that is, 10 yards. 6. 4 times 2 are S, and 4 times are, or 27, which added to 8 make 10 bushels. 12. It would take 1 man 3 times as long as it would 3 men. Ans. 133 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. 384 rods. 15. Ans. 12 yards. 7 is 3, or 13. must be found first, much. of 7 is 3, This manner of reasoning may be applied to any number. To find 4 of 38: it is 38, for 4 of 38 is 38 times as much as of 1, and 4 of 1 is 4, consequently of 38 is 38, and 33 is 53. 40. To find of a number, and then will be 2 times as and 2 times are, or 43. 74. of 50 is 5o, or 5; † 4 times 5 are 20, 4 times added to 20 make 223. 9 is 4 times as much ; are, or 23, which NOTE. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Ans. 14 apiece. 3. 1 of 3 is ; of a bushel apiece. 4.3 of 7 is 44; he gave away 41 and kept 24. 6. 1 half dollar a yard, or 50 cents. 7. of 7 is 7, or 13; & of a dollar is of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. of 8 is 13. of 100 is 333. Ans. 1 dollar and 332 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 5 dollars. Ans. 5 dollars and 2 shillings, or 333 cents. 13. If 7 pounds cost 40 cents, 1 will cost 5 cents; 10 pounds will cost 574 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; 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 1 and 1 half, which is 3. 15. 42 is of 5 times 4 and 4, which is 224. B. 4. 5 is 3 times of 5, which is §, or 13. 30. If 8 is of some number, of 8 is of the same number. of 8 is 23, 24 is 4 of 4 times 23, which is 103; therefore 8 is 40. If 8 is, of 8 is ; or 93; therefore 8 is of 9 2. of 10%. of 8 is 3, is of Y, 52. If of a ton cost 23 dollars, † of a ton must be of 23, that is 43 dollars, and the whole would P cost 9 times as much, that is, 413. 69. 1 of 65 is 73; 74 is of 5 times 7%, which 361. 65 is of 361. C. 4. 37 is & of 32§, which taken from 37 leaves 4. Ans. 4 dollars. 5. 7 feet must be of the whole pole. 6. If he lost, he must have sold it for 7 of what it cost. 47 is 7 of 603. Ans. 60 dollars and 42§ cents. Miscellaneous Examples. 1. The shadow of the staff is 4 of the length of the staff; therefore the shadow of the pole is of the length of the pole. 67 is of 833. Ans. 837 feet. 2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7; 7 of 60 minutes |