times 3 are 18. Consequently, he will see 18 divided into 6 equal parts. It will be well to let the pupil prove a large number of the examples on the plate. The pupil will be very likely to say 3 is the 6th part of 18, because 3 times 6 are 18. Be careful to make him say it the other way, viz. 6 times 3 are 18. 14. I third of 9 is 3; is 2 times as much as įs therefore of 9 is 6. 19. 1 barrel will cost part of 12 dollars ; 3 barrels will cost of 12 dollars. 7 barrels will cost / of 12 dollars. 37. What is of 32 of 32 is 4, are 5 times 4, or 20. B. 11. { of 20 18 4; } are 7 times 4, or 28; and 28 is 4 times 6, and of 6. C. 3. 1 half of 10 is 5, of 10 are 4; 5 and 4 are 9. He gave away 9 and had 1 left. 4. 1 yard will cost of what 3 yards cost. of 6 dollars is 2 dollars. 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 6. 3 apples will cost of what 9 cost; or 6 cents. 7. 2 is of 3; therefore 2'oranges will cost of what 3 cost. of 18 cents are 12 cents. 8. of 25 are 20. The 10 apples cost 20 cents, which was 2 cents apiece. 11. of 42 are 12, and 6 times 12 are 72. 72 dollars. 13. 3 is of 4. of 12 dollars are 9 dollars. Or 4 yards at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards. 14. Solved like the 13th. Ans. 15 cents. 15. Since 1 is of 3, 7 is of 3. 1 of 15 cents are 35 cents. Or, 3 oranges al 15 cents, is 5 cents apiece : 7 times 5 are 35 cents. Note. In questions of this kind it is generally the simplest way to find what 1 article will cost, then it may easily be told how much any number will cost. 19. 4 men would do it in 1 half the time that 2 would do it. Or, you may say, if 2 men would do it in 6 days, 1 man would do it in 12 days, and 4 men in of that time, or 3 days. SECTION VI. A. 4. 2 halves of any number make the whole number. Therefore 2 is 1 half of 2 times 2; or 4. It is į of 4 times 2, or 8. Let the pupil answer these questions in the following manner: 4 is į of 3 times 4; 3 times 4 are 12. 5 is of 7 times 5; 7 times 5 are 35. B. 2. 4 is 2 times 2. 4. 6 is 2 times 3. 16. 2 thirds of any number is twice as much as I of the same number. If 4 is of some number, then 1 half of 4 or 2 is } of that number; 2 is of 6: therefore 4 is of 6. 20. If 6 is į of a number, ļof 6 or 2 is of the same number ; 2 is į of 8; therefore 6 is of 8. 23. It is evident that of a pound will cost only of what will cost. If cost 6 cents, 4 will cost 2 cents, and the whole pound 14 cents. 26. It will probably be perceived by this time, that of a number being given, it is necessary to find 4, and then the number is easily found; 4 being , 2 is +, and 2 is of 14. 45. 24 being ), 1 of 24 or 3 will be $; 3 is 4 of 27. C. 6. 20 being 4,5 is t, and 5 is 4 of 35; and 35 is 5 times 6, and of 6. ! D. 4. 18 is 3 times 6, and 6 is į of 4 times 6, or 24. Ans. 24 dollars. 6. 54 is of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars. 7. 10 feet is of 15 feet. 8. If are under water, there must be į out of the water. 4 is į of 12. 9. Iff are under water there must be out of the water. 6 is į of 10. 10. # and 4 are . bear cherries and peaches; consequently, the 10 which bear plums must be the other; 10 is 3 of 35. 10 bear peaches' and 15 bear cherries. 11. á, and s, and , and }, are 7; therefore 12 must be the other of the whole. The whole number is 54. Miscellaneous Examples. 6. The grey-hound gains upon the fox 4 rods in a minute. It will take him 20 minutes to gain 80 rods. 8. of 24. Or you may say, 1 sheep would cost 3 dollars, and 3 sheep 9 dollars. 9. 30 horses will eat 10 times as much as 3 horses. 11. 10 dollars apiece, and 2 dollars a yard. 12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for 5 weeks. 14. It would take them 5 times as long to eat 40 bushels, as it would to eat 8 bushele. • 15. 4 horses would eat 4. bushels in 3 days, and it would take them 9 times as long to eat 36 busholsi Ans. 27 days. 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 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 VIL. A. 13. of 20 is 4, $ are 16; 16 being 4, 2 ist 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 f of 48) and 48 is 6 times 7 and of 7. B. 1. of 15 are 12; 12 is 6 times 2 ; 2 is in of 20 (12 is to of 20); } of 21 is 7; 20 is 2 times 7 and # of 7. 2. of 18 are 24 ; 24 is of 27; 4 of 35 is 5; 27 is 5 times 5 and 4 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 VIII. Explonation 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 vertiral 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 |