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?

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?

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.

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 3 of a dol lar per yard; what did he receive for it?

2. What is 5 times ? Ans. × 5=§=13.
3. How many are 4 × 3?

4. How many are 3 × 4?
5. How many are 3 times

× 5 ?

× 7 ?

?

Ans. Three times & are §, or 3.

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 or 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 18? 9 times 17?

14. How many are 6 times

15. How many are 5 times ?

Ans. Five times are, or 3.

? 11 times ?

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 13? 49 times 19. What cost 8 boxes of strawberries, at dollar per box?

20. What cost 25 yards of sheeting, at lar per yard?

?

of a

of a dol

21. If it costs 13 of a dollar to build a rod of wall, what will it cost to build 20 rods?

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; 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, as it is that of 6 cents is 3 cents. Hence, dividing the numerator of a fraction by any number, is dividing the fraction by the same number.

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. ÷3=4.

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?

30. What is of 14?

Ans. 1÷7=&. 31. What is ÷ 2? 14 ÷ 4? ÷ 5?

32. What is ÷ 5? 24÷6? 42÷8?

33. Hannah divided one third of a pie equally be tween 2 children; what part of the whole pie did she give to each child?

Solution. One third equals, and

is, ofis ; and therefore she gave pie to each ?

NOTE. Multiplying the denominator by 2 makes twice as many parts in the unit, and therefore each part is only one half as great; hence, multiplying the deno.nirator by any number, divides the fraction by the same number.

34. Three fifths of a barrel of flour were divided equally among 4 poor families: : what part of a barrel

did each receive?

35. If of a bushel of chestnuts are divided equally

among 5 boys, what part of a bushel does each receive?

36. What is 1 sixth of? 1 eighth of ?

37. What is 1 ninth of ? 1 twelfth of 11?

용

38. Mary having a nice large pineapple, gave

of it to Sarah, to Fanny,

to Laura, and kepi

the rest herself; what part of the pineapple did she give away? What part did she keep?

39. What is +A+A?

40. What is 117? H-A? A +✩? 41. What is + 15 + 15 +1%? 18 — 11

42. What is

LESSON II.

of a whola

1. ONE half is equal to how many sixths? Solution. Since & make a whole one, one will be of, which is ; therefore, etc.

2. One third of an apple is equal to how many sixths of an apple?

3. James gave of a pear to William, to George, and kept the rest himself; how much did he give away? How much did he keep?

NOTE. Change the fractions to sixths. Before fractions that have different denominators can be added together, or before one can be subtracted from another, it is necessary to change them to other fractions that have like denominators.

4.

what? 1--what?

=

5. A man gave of a dollar to one of his workmen, and 2 of a dollar to another; how many fourths of a dollar did he give to both? 6. are how many?

How many dollars?
How many times 1?

NOTE. Read the first question in Ex. 6 as follows: One half and three fourths are how many fourths? Read all similar examples in like manner.

7. A man gave of a barrel of apples to one of his neighbors, and of a barrel to another; did he give more to one than to the other?

8. and are how many? How many times 1 ? 9. Henry gave of an orange to his brother John, to his sister Marion, and kept the rest himself. With this division the selfish John was dissatisfied, saying that Henry had given more to Marion than to him. No, John, said Marion, Henry has given just as much to you as to me. Now which was right, John or Marion? What part of the orange did Henry keep for himself?

10.

is how many?

11. }+&+}=how many? How many times 1? 12. Samuel worked of a day for Mr. Adams, and of a day for Mr. Daniels. Mr. Adams paid Samuel 50 cents. Now, at the same rate, what should Mr. Daniels pay him? How much, at that rate, will Samuel earn in a day? What part of a dollar?

13. are how many?

14. A boy wished to give of a pear to his sister, and to his brother; and in order to do this most conveniently, he first cut the pear into 6 equal pieces; how many pieces did he give to each? How many

pieces had he remaining?

15. Samuel gave of a dollar to one poor boy, and of a dollar to another; what part of a dollar did he give away? How much more to one boy than to the other?