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 is of the length cf the staff, therefore the shadow of the pole must be the length of the pole. 18 feet is 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 must 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 firkin. 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, are 16; 16 being \, 2 is 4 2 is of 14, and 16 is of 14. 16. of 28 are 12; 12 is 2 times 6, and 6 is of 4€, (12 is of 48 and 48 is 6 times 7 and of 7. B. 1. of 15 are 12 ; 12 is 6 times ?; 2 is to of 20 (12 is is of 20); of 21 is 7; 20 is 2 times 7 and 4 of 7. 2. 4 of 18 are 24 ; 24 is of 27; of 35 iz 5; 27 is 5 times 5 and 4 of 5. ? C. This article contains the multipiication 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 divisiblo 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 de 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 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 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 ose parts are to 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 seco ond 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 id 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 . SECTION IX. 1. 2. signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or ģ. 6. 7 times į is į, or 2. 5 times are 'which takes 3 squares and 1 space. Ans. 3j. 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. 2. 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 , or throu whole ones, which added to 12 make 15. 32. 2 times :3 are 6, and 2 times are q, which added to 6 make 64. 40. 10 barrels of cider at 3. dollars and a barael; 10 barrels at 3 dollars, would be 30 dollars, then 10 times 5 is 5, or 8 and of a dollar. Ans. 387 dollars. C. 2. to each would be 3 times , or, which are 21 oranges. 3. or 2 bushels. 4. 7 times are 4, or 5 gallons. 5. 8 yards and for 2 yards, that is, 10 yards. 6. 4 times 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. 38, rods 15. Ans. 12 yards. SECTION X. A. 21. į of 1 is į į of 2 is 2 times as much or f. of 4 is , or 1}. { of 5 is , or 13. of 6. 18 g, or 2. į of 7 is ž, or 2z. 27. } of 1 is : { of 2 is g. of 3 is . o 7 is }, or 1 This manner of reasoning may be applied to any number. To find of 38: it is ¥, for of 38 is 38 times as much as of 1, and of 1 is , consequent sy of 38 is 39, and xs is 57. 40. To find of a number, į must be found first, and then will be 2 times as much. of 7 is }, and 2 times are 1, or 4. 74. 4 of 50 is , or 5% ; $is 4 times as much 4 times 5 are 20, 4 times are », or 23 whi added to 20 make 22. NOTE. The manner employed in example 40th is best for small numbers, and that in the 74th for large nambers. B. 2. Ans. 19 apiece. 7. } of 7 is 7, or 15 ; of a dollar is f of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. ☆ of 8 is 15. of 100 is 332. Ans. I 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 54 dollars. Ans. 5 dollars and 2 shillings, or 33; centk. 13. If 7 pounds cost 40 cents, 1 will cost bouf cents; 10 pounds will cost 574 cents. 16. I cock would empty it in 6 hours, and 7 cocks |