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, off from the dividend, will form the true remainder.

EXAMPLES FOR PRACTICE.

cut

5. Divide 13112297 by 8900.

497654325

306787

6. Divide 71897654325 by 700000000.

7. Divide 3456789123456787 by 990000.

8. Divide 967231731328000 by 1020000000. 9. Divide 33166405115000 by 1600000000. 10. Divide 18191618562300 by 1000000000.

411328000

5115000

618562300

11. Divide 4766666000000 by 55550000000. 44916000000

VI. QUESTIONS INVOLVING FRACTIONS.

ART. 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.

2. When any number or thing is divided into three equal parts, one of the parts is called one third (1); 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 (3).

4. When any number or thing is divided into five equal parts, one of the parts is called one fifth (1); two parts, two fifths (?); three parts, three fifths (3); and four parts, four fifths (†).

QUESTIONS. Art. 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 And 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 32 ?

Of 8? Of 16? Of 20? Of 28?

9. What is 1 third of 9? Of 12? Of 15? Of 27? Of 30? Of 36? Of 60 ?

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 (g) less 3

fifths (3)

= 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 (3) to his son, and 3 ninths (3) to his daughter, he gave them both + 3

; and

since there are 9 ninths (3) in the farm, he must have given his

wife

Ans..

21. In a certain school

of the pupils study grammar, study arithmetic, geography, and the remainder philosophy. 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, what cost of a yard? What cost of a yard?

ILLUSTRATION. -If 1 yard cost $8, 4 of a yard will cost of $8 = $2; and if of a yard cost $2, will cost three times as = $6.

much; 3 times $2

Ans. $6.

24. If an acre of land cost $24, 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 barrel?

ILLUSTRATION.

dollars = $8.

If cost 2 dollars, will cost 4 times 2
Ans. $8.

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, what will a hogshead cost?

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

ILLUSTRATION. If cost $21, will cost of $21, and ↓ of $21 is $3; and will cost 8 times $3 = $24, and 10 acres will = $240. Ans. $240.

cost 10 times $24:

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

31. If of a barrel of apples cost $1.50, what costs a barrel? What cost 10 barrels ?

32. When $49 are paid for 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 = 4 halves, and the 1 half added makes 5 halves. Ans.. 34. How many half-bushels in 44 bushels of oats? In 51 bushels? In 7 bushels? In 9 bushels?

35. How many eighths of a dollar in 2 dollars? In 43 dollars? In 7g 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 5,7%

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 halves (§) as 2 is contained times in 6. 2 is contained in 6, 3 times.

Ans. 3 barrels.

38. How many firkins of butter in § firkins? 39. How many whole numbers in 10? In 15? 40. How many whole numbers in 16? In ? 48 ?

In 20 firkins?
In 45?
In 32? In

41. If a skein of silk is worth 3 cents, what are 6 skeins worth?

what must I For 12 pairs?

pay for

42. Bought one pair of boots for $61; 4 pairs? For 8 pairs? For 10 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 4 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 and of 12; 4 times 12 are 48, and of 12 is 6; 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 9 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 cents, what is the value of 16 yards? 12 yards? 8 yards? 11 yards?

1 bushels

bushels.

Now, if

48. If 14 bushels of corn cost $1.20, what will 1 bushel cost? ILLUSTRATION. cost $1.20, will cost of $1.20 $0.40; and or a whole bushel will cost 2 times $0.40 $0.80.

Ans. $0.80.

49. If 23 pounds of coffee cost 60 cents, what will 1 pound cost?

12 pounds. If 12 cost 60 5 cents; and g, or a pound,

will cost 5 times 5 cents = 25 cents.

50. How many times will 60 contain 2?

Ans. $0.25.

51. Paid $54 for 71⁄2 barrels of oil; what cost 1 barrel?

52. How many times is 7 contained in 54 ?

Ans. $7

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

54. How many times will 66 contain 5?

55. Gave $40 for 63 yards of broadcloth; what cost 1 yard? 56. How many times is 63 contained in 40?

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

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.

QUESTIONS. Art. 61. What is the rule for multiplying by 25? What is the reason for the rule?