16. If 2 men spend 12 dollars in 1 week, 1 man will spend 6 dollars in 1 week, and 30 dollars in 5 weeks, and 3 men would spend 3 times as much, or 90 dollars. 17. The shadow of the staff is of the length of the staff, therefore the shadow of the pole must be the length of the pole. 18 feet is f of 27 feet. 20. It would take 2 men 3 times as long to do it as it would 6 men. 23. 8 men would do a piece of work 1 half as large in 2 days, and it would take 2 men 4 times as long to do it, or 8 days. 28. He inust sell it for 56 dollars in order to gain 16 dollars. 56 dollars is 7 dollars per barrel. 29. It cost him 35 dollars, and he must sell it for 45 to gain 10 dollars ; 45 dollars is 9 dollars a firking 30. Ans. 56 cents, see section VI. 33. If it would last 3 men 10 months, it would last 1 man 30 months, and 5 men 6 months. 34. There are 8 times 5 in 40, and since the other would build as many times 9, as the first does 5, he would build 8 times 9 or 72 rods, SECTION VII. A. 13. of 20 is 4, 4 are 16; 16 being 4,2 is; 2 is 4 of 14, and 16 is of 14. 16. of 28 are 12; 12 is 2 times 6, and 6 is ; of 48, (12 is of 48) and 48 is 6 times 7 and 4 of 7. B. 1. f of 15 are 12; 12 is 6 times 2; 2 is i'o of 20 (12 is 1 of 20); of 21 is 7; 20 is 2 timos 7 and of 7, 2. of 18 are 24 ; 24 is f 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 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 sumns 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, cach 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, Ist, 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 manner 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 ad 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 s. SECTION IX. A. 2. į signifies that 1 thing is divided into 3 equal parts, and I part taken. Therefore 2 times I third is 2 parts, or . 6. 7 times į is j, or ?. 10. On the plate in the 3d row,5 times are , which takes 3 squares and I space. Ans. 3}. 24. In the 9th row take 4 spaces or 9ths, and repeat them 5 times, which will make y, and will iake 2 squares and 2 spaces. Ans. 2. B. 4. 4 times 2 are 8, and 4 times 1 half are 4 lialves, or 2, which added to 8 make 10. 18. 4 times 3 are 12, and 4 times į are , or 1hree whole ones, which added to 12 make 15. 32. 2 times 3 are 6, and 2 times are y, which added to 6 make 69. 40. 10 barrels of cider at 3 dollars and j a barrel; 10 burrels at 3 dollars, would be 30 dollars, then 10 times is 50, or 8 and į of a dollar. Ans. 331 dollars. C. 2. to each would be 3 times ], or f, which oranges. 3. ' or 2 bushels. 4. 7 times are *, or 5} gallons. 5. 8 yards and or 2 yards, that is, 10 yards. 6. 4 tirnes 2 are 3, and 4 times are ', or 2}, 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 1 man; and in 4 days they would build 4 times as much as in 1 day. Ans. 38rods. 15. Ans. 12 yards. . are 4. of 7 is 4'}; he gave away 41 and kept 2 SECTION X. A. 21. of 1 is į į of 2 is 2 times as much, or 5 of 4 is f, or lī. { of 5 is , or 13. j of 6 is f, or 2. of 7 is j, or 2j. 27. į of 1 is š. } of 2 is . } of 3 is š. of 7 is j, or lý. This manner of reasoning may be applied to any number. To find į of 38: it is , for 4 of 38 is 38 times as much as 4 of 1, and of 1 is 4, consequently 4 of 38 is 349, and 48 is 5. 40. To find of a number, į must be found first, and then j will be 2 times as much. } of 7 is }, and 2 times į are 1, or 43. 74.. ☆ of 50 is 5, or 5 ; 4 is 4 times as much ; 4 times 5 are 20, 4 times are in, or 25, 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. 1$ apiece. 6. i half dollar a yard, or 50 cents. 7. } of 7 is }, or lz; of a dollar is ş of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. $ of 8 is 13. 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 5.1 dollars. Ans. 5 dollars and 2 shillings, or 33; cents. 13. If 7 pounds cost 40 cents, I will cost 5 cents; 10 pounds will cost 57 cents. 16. I cock would empty it in 6 hours, and 7 cock |