26. A man had 36 sheep, and sold 5 ninths of them at 4 dollars a head; how many dollars did he receive? 27. How many are 4 times 5 ninths of 36? 28. If 5 eighths of a box of raisins cost 15 shillings, how many bushels of oats, at 4 shillings a bushel, will pay for a box of raisins? 29. 8 fifths of fifteen are how many times 4? 6? 30. A man gave 25 cents for his breakfast, which was 5 eighths of what he gave for his dinner; what did he give for his dinner? 31. A man paid 45 cents for his dinner, which was 9 sevenths of what he paid for his breakfast; what · did his breakfast cost? 32. A teacher, when asked how many pupils he had, replied that they were in 2 rooms: in one room there were 32, and 5 eighths of these were fifths of the number in the other; how many pupils were there in the other room? How many in both? 33. 5 eighths of 32 are 2 fifths of what number? 34. A boy being asked how old his dog was, said that his cat was 8 years old, that the age of his cat was 4 thirds of the age of his rabbit, and that his dog was twice as old as his rabbit; how old was his dog? LESSON XII. 1. How many are 2 times 13? 2. How many are 2 times 14? 3. How many are 2 times 15? 4. How many are 2 times 16? 5. How many are 2 times 17? 6. How many are 2 times 18? 7. How many are 2 times 19? 8. How many are 2 times 20? 9. How many are 2 times 21? 10. How many are 2 times 22? 11. How many are 2 times 23? 13 times 2? 14 times 2? 15 times 2? 16 times 2? 17 times 2? 18 times 2? 19 times 2? 20 times 2? 21 times 2? 22 times 2? 12. How many are 2 times 24? 13. How many are 2 times 25? 14. How many are 3 times 13? 15. How many are 3 times 14? 16. How many are 3 times 15? 17. How many are 3 times 16? 18. How many are 3 times 17? 19. How many are 3 times 18? 20. How many are 3 times 19? 21. How many are 3 times 20? 22. How many are 3 times 21? 23. How many are 3 times 22? 24. How many are 3 times 23? 25. How many are 3 times 24? 26. How many are 3 times 25? 27. How many are 4 times 13? How many are 4 times 14? 29. How many are 4 times 15? 30. How many are 4 times 16? 31. How many are 4 times 17? 32. How many are 4 times 18? 33. How many are 4 times 19? 34. How many are 4 times 20? 35. How many are 4 times 21? 36. How many are 4 times 22? 37. How many are 4 times 23? 38. How many are 4 times 24? 39. How many are 4 times 25? 40. How many are 5 times 13? 41. How many are 5 times 14? 42. How many are 5 times 15? 43. How many are 5 times 16? 44. How many are 5 times 17? 45. How many are 5 times 18? 46. How many are 5 times 19? 47. How many are 5 times 20? 48. How many are 5 times 21? 24 times 2? 25 times 2? 13 times 3? 14 times 3? 15 times 3? 16 times 3? 17 times 3? 18 times 3? 19 times 3? 20 times 3? 21 times 3? 22 times 3? 23 times 3? 24 times 3? 25 times 3? 13 times 4? 14 times 4? 15 times 4? 16 times 4? 17 times 4? 18 times 4? 19 times 4? 20 times 4? 21 times 4? 22 times 4? 23 times 4? 24 times 4? 25 times 4? 13 times 5? 14 times 5? 15 times 5? 16 times 5? 17 times 5? 18 times 5? 19 times 5? 20 times 5? 21 times 5? 49. How many are 5 times 22? 50. How many are 5 times 23? 51. How many are 5 times 24? 52. How many are 5 times 25? 53. How many are 6 times 13? 54. How many are 6 times 14? 55. How many are 6 times 15? 56. How many are 6 times 16? 57. How many are 6 times 17? 58. How many are 6 times 18? 59. How many are 6 times 19? 60. How many are 6 times 20? 61. How many are 6 times 21? 62. How many are 6 times 22? 63. How many are 6 times 23? 64. How many are 6 times 24? 65. How many are 6 times 25? 66. Count by 6's to 96; thus, 6, 12, 18, etc. 67. Count backward by 6's; thus, 96, 90, 84, etc. 68. Count by 6's from 3 to 99; thus, 3, 9, 15, etc. 69. Count backward by 6's; thus, 99, 93, 87, etc. 70. Count by 7's to 98; thus, 7, 14, 21, etc. 71. Count backward by 7's; thus, 98, 91, 84, etc. 22 times 5? 23 times 5? 24 times 5? 25 times 5? 13 times 6? 14 times 6? 15 times 6? 16 times 6? 17 times 6? 18 times 6? 19 times 6? 20 times 6? 21 times 6? 22 times 6? 23 times 6? 24 times 6? 25 times 6? NOTE. Such exercises as the last six examples above should be continued only a very few minutes at a time. Let the exercise be varied by allowing the class to recite sometimes in concert, sometimes individually; by allowing one pupil to name one number. and another the next; by letting two or more pupils recite simultaneously. one counting from one number, and another from another number; or by any other mode which shall secure the attention and awaken the interest of the class. The more complicated of the above plans, if adopted. should be commenced with the 2's, the 5's, the 10's, and other numbers which give the.more simple combinations; and they should not be continued unless the pupils have the ability to proceed without confusing each other These exercises, if judiciously conducted, are very valuable; and when the pupil shall have acquired the ability to make all such combinations with accuracy and rapidity, he will have very great facility in all the processes of addition and subtraction. SECTION SEVENTH. LESSON I. 1. A MERCHANT sold 5 yards of cloth, at of a dollar per yard; what did he receive for it? 2. What is 5 times ? Ans. × 5=3=13. 3. How many are × 5 ? × 7? 4. How many are ? X3? × 4? NOTE. The are obtained by multiplying the numerator by 3, and the are obtained by dividing the denominator by 3. In examples like this, it is better to divide the denominator, because it gives the answer in smaller cr lower terms. 6. How many are 5 times? Why? 7. At of a dollar a pound, what will 5 pounds of butter cost? Solution. Five pounds will cost 5 times as much as 1 pound; 5 times of a dollar are of a dollar — $13, 8. At of a dollar per pound, what will 4 pounds of butter cost? 9. What will 6 yards of silk cost, at 1 of a dollar per yard? 10. If 1 horse eats how much will 7 horses of a ton of hay in a month, eat in the same time? 11. If 1 man can reap of an acre of rye in a day, how much can 8 men reap in the same time? 12. If 1 yard of cloth costs 1 of a dollar, what will 8 yards cost? 13. How many are 8 times ? 9 times ? 11 times Ans. Five times are †, or 3. ? ? NOTE. The pupil will observe that if a fraction is multiplied by its denominator, the product is the numerator. 16. How many are 11 times? Why? 17. How many are 9 times ? 15 times 7? 18. How many are 23 times ? 49 times ? 19. What cost 8 boxes of strawberries, at § of a dollar per box? 20. What cost 25 yards of sheeting, at of a dollar per yard? 21. If it costs what will it cost to build 20 rods? of a dollar to build a rod of wall, 22. If a boy runs 1 of a rod in 1 second, how many rods, at the same rate, will he run in 15 seconds? 23. If a locomotive runs of a mile in 1 minute, how far will it run in 75 minutes? 24. If 2 bushels of apples cost of a dollar, what will 1 bushel cost? Solution. One bushel will cost 1 half as much as 2 bushels; 1 half of of a dollar is of a dollar; 9 therefore, if 2 bushels cost of a dollar, 1 bushel will cost of a dollar. NOTE. To find of a number is the same as to divide the number by 2. It is just as evident that of & is 2, as it is that of 6 cents is 3 cents. Hence, dividing the numerator of a frac tion by any number, is dividing the fraction by the same number. 34. 25. If a boy walks 3 miles in of an hour, how long will it take him to walk 1 mile? 26. What is of § ? Ans. 27. If 4 men can reap of an acre of rye in an hour, how much can 1 man reap in the same time? 28. If a boy can run of a mile in 3 minutes, how far can he run in 1 minute? 29. What is 1 dozen of eggs worth, if 7 dozen are worth of a dollar? of 1? Ans. ÷ 7 = 25. 30. What is |