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 4 of 4 times 2, or 8. Let the pupil answer these questions in the following manner: 4 is 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 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 of a number, 5 of 6 or 2 is of the samo number; 2 is of 8; therefore 6 is of 8. 23. It is evident that of a pound will cost only 3 of what will cost. If cost 6 cents, — will cost 2 cents, and the whole pound 14 cents. 26. It will probably be perceived by this time, that of & number being given, it is necessary to find , and then the number is easily found ; 4 being 7, 2 is t, and 2 is of 14. 45. 24 being , 5 of 24 or 3 will be }; 3 is 3 of 27. C. 6. 20 being 4, 5 is !, and 5 is of 35; and 35 is 5 times 6, and of 6. D. 4. 18 is 3 times 6, and 6 is 1 of 4 times 6, or 24. Ans. 24 dollars. 6. 54 is of 48; 12 yards at 48 dollarg is 4 dollars a yard. He gained 6 dollars. 7. 10 feet is of 15 feet 8. If g are under water, there must be s out of the wa ter. 4 is of 12. 9. If are under water, there must be out of the war ter. 6 is of 10. 10. $ and 8 are & bear cherries and peaches; consequently, the 10 which bear plums must be the other &; 10 is of 35. 10 bear peaches, and 15 bear cherries. 11. $, and $, and }, and }, are }; therefore 12 must be ihe other of the whole. The whole number is 54. Miscellaneous Examples. 6. The greyhound gains upon the fox 4 rods in a minate. It will take bim 20 minutes to gain 80 rods. 8. f 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 bushels. 15. 4 horses would eat 4 bushels in 3 days, and it would take them 9 times as long to eat 36 bushels. 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 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, anc he must sell it for 45 to gain 10 dollars; 45 dollars is 9 dollars a firkin. 30. Ans. 56 cents, see section VI. men. 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 4, 2 is ; 2 is 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. $ of 15 are 12; 12 is 6 times 2; 2 is to of 20 (12 is 8 of 20); } of 21 is 7; 20 is 2 times 7 and of 7. 2. of 18 are 24; 24 is $ of 27; 1 of 35 is 5; 27 is 6 times 5 and f 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. A. 1. In one there are 2 halves; in 2 there are twice as many halves, that is four halves or 4. 2. In one there are 2 halves, in 3 there are three times two halves, that is, 6 halves, or f. 3. In one there are 3 thirds, in 2 there are twice 3 thirds, that is, 6 thirds, or f. 15. Draw two lines on the board thus Each lina call one ; divide the upper line into two equal parts, each part is one half of one, or one half; divide the lower line in the same way, and then call for the answer. The form of the question may be varied by asking how many half apples there are in two apples. 37. In 5 and 2 thirds, how many thirds ? Draw five parallel equal lines, divide them into thirds, and draw another parallel with the others and two thirds as long, let the scholar count the thirds. In all these cases of illustration on the board, it is presumed that the scholar will soon take the bint, and devise illustrations for himself at his seat. SECTION IX. A. 2. į signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or . 6. '7 times ] is , or 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 les 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 68. 40. 10 barrels at 3 dollars and a barrel ; 10 barrels at 3 dollars; would be 30 dollars, then 10 times á is 60, or 8 and 4 of a dollar. Ans. 384 dollars. C. 2. I to each would be 3 times t, or, which are 24 oranges. 3. 1 or 2 bushels. which added to 8 make 104 bushels. 12. It would take 1 man 3 times as long as it would 3 men. Ans. 134 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. 38f rods. 15. Ars. 12 yards. SECTION X. A. 21. of 1 is to of 2 is 2 times as much, or . of 4 is t, or 13. of 5 is f, or 15. of 6 is f, or ž. of 7 is , or 27. 27. 1 of 1 is . 1 of 2 is . } of 3 is . of 7 is 7, or 1ę. This manner of reasoning may be applied to any number. To find of 88: it is 2, for of 38 is 38 times as much as of 1, and 4 of 1 is 4, consequently of 89 is , and 38 is 5%. 40. To find f of a number, $ must be found first, and then will be 3 times as much. of 7 is , and 2 times fare *, or 45 74. of 50 is one, or 56; $is 4 times as much; 4 times 5 are 20, 4 times are 2, or 2%, which added to 20 make 223. Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Ans. 1apiece. 3. 1 of 3 is ; fe of a bushel apiece. 4. of 7 is 43; he gave away 41, and kept 24. 6. i half dollar a yard, or 50 cents. 7. } of 7 is }, or 14 ; of a dollar is of 100 cents, which is 40 cents. Ans. i dollar and 40 cents a bushel. 8. of 8 is 14. of 100 is 33. Ans. 1 dollar and 334 cents, or it is 1 dollar and 2 shillings. 9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and , and 2 bushels will cost 51 dollars. Ans. 5 dollars and 2 shillings, or 33 cents. 13. If 7 pounds cost 40 cents, 1 will cost 55 cents; 10 pounds will cost 57 cents. 16. 1 cock would empty it in 6 hours, and 7 cocks would empty it in 4 of 6 hours, or 4 of 1 hour, which is of 60 minutes; 4 of 60 minutes is 51$ minutes. conso SECTION XI. A. 2. 2 halves of a number make the number quently 1 and 1 half is the half of 2 times 1 an 1 half, which is 3. 15. 44 is of 5 times 4 and 4, which is 22%. 17. 44 is of 9 times 44, which is 39%. B. 4. 5 is 3 times ] of 5, which is , or 14. 30. If 8 is of some number, 1 of 8 is 1 of he same number. 3 of 8 is 23, 24 is of 4 times 23 which is 10%; therefore 8 is f of 103: 40. If 8 is f. 2 of 8 isti 1 of 8 is $, & is of the, or 94 ; therefore 8 is of 98. 52. If of a ton cost 23 dollars, $ of a ton must be of |