NOTE. When by the operation there is a last remainder, to it must be annexed the figures cut off from the dividend to form the true remainder. Should there be no last remainder, then the significant figures, if any, cut off from the dividend, will form the true remainder.

7. Divide 3456789123456787 by 990000. 8. Divide 967231731328000 by 1020000000. 9. Divide 33166405115000 by 1600000000. 10. Divide 18191618562300 by 1000000000. 11. Divide 4766666000000 by 55550000000.

QUESTIONS INVOLVING FRACTIONS.

60. IF a unit or individual thing is divided into equal parts, each of the parts is called a Fraction of the number or thing divided. Hence

A Fraction is one or more equal parts of a unit.

ILLUSTRATIONS.-1. When any number or thing is divided into two equal parts, one of the parts is called one half, and is written thus:

2. When any number or thing is divided into three equal parts, one of the parts is called one third (†); two of the parts are called two thirds (3).

3. When any number or thing is divided into four equal parts, one of the parts is called one fourth (4); three of the parts, three fourths ($).

4. When any number or thing is divided into five equal parts, one of the parts is called one fifth (); two parts, two fifths (); three parts, three fifths (); and four parts, four fifths (4).

60. What is a fraction? What is meant by one half of any number or thing? How is it written What is meant by one third, and how is it written? What by one fourth, and how written? What by one fifth, and how written? What by four fifths, and how written? How do you find one half of any number? How one third ? How one fourth? &c. How many halves make a whole one? How many thirds? How many fourths? How many fifths?

5. When any number or thing is divided into six equal parts what is one of the parts called? Two parts? Five parts?

6. When a number or thing is divided into 7 equal parts, what is 1 part called? 2 parts? 3 parts? 4 parts? 5 parts? 6 parts?

7. When a number or thing is divided into 9 equal parts, what is 1 part called? 2 parts? 4 parts? 5 parts? 7 parts? 8 parts?

8. What is 1 half of 4? Of 8? Of 16? Of 20? Of 28 ? Of 32?

9. What is 1 third of 9? Of 12? Of 15? Of 27 ? Of 36? Of 60?

Of 30?

10. What is 1 fourth of 8? Of 16? Of 20? Of 24? Of 40? Of 48? Of 100?

11. What is 1 fifth of 10? Of 25? Of 30? Of 35? Of 45? Of 50? Of 55? Of 65?

12. What is 1 sixth of 12? Of 18? Of 30? Of 42? Of 60? Of 72? Of 90?

13. How many fourths in 1 apple?

14. How many fourths in 2 apples? In 3 apples? In 8 apples? In 16 apples?

15. How many fifths in 1 barrel of flour? In 3 barrels? In 5 barrels? In 7 barrels? In 9 barrels ?

16. How many sixths in 1 bushel of wheat? In 4 bushels ? In 7 bushels? In 9 bushels? In 12 bushels?

17. James owns 3 fifths of a kite, and his brother Thomas the remainder. How many fifths does Thomas own?

ILLUSTRATION. - Since there are 5 fifths in the kite, if James owns 3 fifths, there will remain for Thomas 5 fifths (§) less 3 fifths 2 fifths.

()

Ans. 2 fifths.

18. From a load of hay I sold 4 sevenths; how many sevenths remain ?

19. John Jones found a large sum of money; he gave 5 eighths of it to the poor of the parish; how much did he reserve for himself?

20. John Smith gave 2 ninths of his farm to his son, 3 ninths to his daughter, and the remainder to his wife; how many ninths did his wife receive?

ILLUSTRATION. Since he gave 2 ninths () to his son, and 3 ninths () to his daughter, he gave them both +; and since there are 9 ninths () in the farm, his wife must have received -- †.

Ans. .

21. In a certain school of the pupils study grammar, geography, and the remainder philosophy.

study arithmetic,

What part of the school study philosophy?

22. J. Dow spends of his time in reading, & in labor, and in visiting. How large a portion of his time remains for eating and sleeping?

23. If a yard of cloth cost 8 dollars, what cost of a yard? What cost of a yard?

ILLUSTRATION.

8 dollars

If 1 yard cost 8 dollars, of a yard will cost of 2 dollars; and if of a yard cost 2 dollars, will cost three times as much, or 6 dollars. Ans. 6 dollars.

=

24. If an acre of land cost 24 dollars, what will of an acre cost? What will cost?

25. When 96 cents are paid for a bushel of rye, what cost → of a bushel?

26. If of a barrel of flour cost 2 dollars, what cost of a barrel?

=

ILLUSTRATION. - If cost 2 dollars, will cost 4 times 2 dollars 8 dollars. Ans. 8 dollars.

27. If of an acre of land cost 24 dollars, what will of an acre cost?

28. If of a hogshead of molasses cost 11 dollars, what will a hogshead cost?

29. If of an acre of land cost 21 dollars, what cost of an ? What cost an acre? What cost 10 acres?

acre

ILLUSTRATION. If cost 21 dollars, will cost of 21 dollars, and of 21 dollars is 3 dollars; and will cost 8 times 3 dollars, or 24 dollars, and 10 acres will cost 10 times 24 dollars, or 240 dollars. Ans. 240 dollars.

30. If of a hogshead of sugar cost 18 dollars, what costs 1 hogshead? What cost 4 hogsheads?

31. If of a barrel of apples cost 150 cents, what costs a barrel? What cost 10 barrels ?

32. When 49 dollars are paid for 7 of a ton of potash, what must be paid for 2 tons?

33. How many half-barrels of flour are there in 2 and a half (21) barrels ?

ILLUSTRATION. Since 1 barrel contains 2 halves, 2 barrels will contain 2 times 2 halves, or 4 halves, and the 1 half added makes 5 halves. Ans.

34. How many half-bushels in 4 bushels of oats? In 5 bushels? In 7 bushels? In 9 bushels?

35. How many eighths of a dollar in 24 dollars? In 48 dollars? In 7 dollars? In 97 dollars? In 12 dollars?

36. How many tenths of an ounce in 4 ounces? In 8 ounces? In 10 ounces?

ounces? In 5%

37. How many barrels of wine in 6 half (§) barrels ?

ILLUSTRATION. Since it takes 2 halves to make one whole one, there will be as many whole barrels in 6 half-barrels as 2 is contained times in 6, or 3 barrels. Ans. 3 barrels.

firkins? In 2o firkins? In 15? In 25?

38. How many firkins of butter in 39. How many whole numbers in 49? 40. How many whole numbers in 6? In ? In 32? In 48 ? 41. If a skein of silk is worth 3 cents, what are 6 skeins worth?

ILLUSTRATION.If 1 skein is worth 3 cents, 6 skeins are worth 6 times as much; 6 times 3 are equal to 6 times 3 and 6 times; 6 times 3 = 18; 6 times=3; 18 cents + 3 cents = 21 cents. Ans. 21 cents.

42. Bought one pair of boots for 6 dollars; what must I pay for 4 pairs? For 8 pairs? For 10 pairs? For 12 pairs?

43. Paid 12 cents for one pound of cloves; what will 6 pounds cost? 10 pounds? 12 pounds?

44. If one pound of butter is worth 12 cents, what are 41 pounds worth?

ILLUSTRATION. If 1 pound is worth 12 cents, 4 pounds are worth 4 times as much; 4 times 12 cents are equal to 4 times 12 cents and of 12 cents; times 12 cents are 48 cents, and 1⁄2 of 12 cents is 6 cents; 48 cents and 6 cents are 54 cents. Ans. 54 cents.

45. When lard is sold for 9 cents per pound, what must be paid for 7 pounds? For 8 pounds? For 94 pounds? 46. Bought 1 pound of coffee at 16 cents; what will 5 pounds cost? 3 pounds? 5 pounds? 6 pounds? 47. If 1 yard of cloth is worth 20 16 yards? 124 yards? 8 yards?

cents, what is the value of 114 yards?

48. If 1 bushels of corn cost 120 cents, what will 1 bushel cost?

13 bushels

=

ILLUSTRATION. bushels. Now, if cost 120 cents, will cost of 120 cents, or 40 cents; and, or a whole bushel, will cost 2 times 40 cents, or 80 cents. Ans. 80 cents.

49. If 2 pounds of coffee cost 60 cents, what will 1 pound cost?

ILLUSTRATION. — 23 pounds = pounds. If cost 60 cents, will cost of 60 cents, or 5 cents; and ğ, or a pound, will cost 5 times 5 cents, or 25 cents. Ans. 25 cents.

50. How many times will 60 contain 22?

rel?

51. Paid 54 dollars for 7

barrels of oil; what cost one barAns. 7 dollars.

contained in 54?

52. How many times is 7

53. How many cords of wood at 5 dollars per cord, can be bought for 66 dollars?

54. How many times will 66 contain 5?

55. Gave forty dollars for 6 yards of broadcloth; what cost 1 yard?

56. How many times is 6 contained in 40?

57. The distance between two places is 110 rods. I wish to divide this distance into spaces of 5 rods each. number of spaces?

Required the

CONTRACTIONS.*

CONTRACTIONS IN MULTIPLICATION.

61. To multiply by 25.

Ex. 1. Multiply 876581 by 25.

OPERATION.

4) 87 658100

219 14 5 2 5 Product. we divide by 4 to obtain the true

RULE.

Ans. 21914525.

We multiply by 100, by annexing two ciphers to the multiplicand; and since 25, the multiplier, is only one fourth of 100, product.

Annex two ciphers to the multiplicand, and divide it by 4.

*If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his progress, they may be omitted for the present, and attended to when he is further advanced.

61. The rule for multiplying by 25? The reason for the rule?