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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 signiticant figures, if any, cut off from the dividend, will form the true remainder.

EXAMPLES FOR PRACTICE.

Quotients.

Rem. 2. Divide 3594 by 80.

44

74 3. Divide 79872 by 240.

332

192 4. Divide 467153 by 700.

667

253 5. Divide 13112297 by 8900.

2597 6. Divide 71897654325 by 700000000.

497654325 7. Divide 3456789123456787 by 990000.

306787 8. Divide 967231731328000 by 1020000000. 411328000 9. Divide 33166405115000 by 1600000000. 5115000 10. Divide 18191618562300 by 1000000000. 618562300 11. Divide 4766666000000 by 55550000000. 44916000000

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 (t); 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 (1); three of the parts, three fourths (4)

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 (*).

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 30 ? Of 36 ? Of 60 ?

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 (1) 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 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 (f) to his son, and 3 pinths (8) 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, i study arithmetic, ii geography, and the remainder philosophy. What part of the school study philosophy ?

22. J. Dow spends 7 of his time in reading, 4 in labor, and 4 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. If 1 yard cost 8 dollars, 4 of a yard will cost of 8 dollars 2 dollars; and if I 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 s of an acre cost ? What will cost ?

25. When 96 cents are paid for a bushel of rye, what cost 11 of a bushel ?

26. If į 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 Z 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 costs of an acre ? What cost an acre ? What cost 10 acres ?

ILLUSTRATION. – If . cost 21 dollars, 1 will cost of 21 dollars, and 4 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 i 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 Yt 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 (24) 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.

34. How many half-bushels in 41 bushels of oats? In 51 bushels ? In 74 bushels ? In 97 bushels ?

Ans. 4.

35. How many eighths of a dollar in 24 dollars ? In 43 dollars? In 7 dollars ? In 97 dollars ? In 12f dollars ?

36. How many tenths of an ounce in 41% ounces? In 510 ounces? In 815 ounces? In 1014 ounces ? 37. How

many barrels of wine in 6 half (s) barrels ? ILLUSTRATION.

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. 38. How

many

firkins of butter in { firkins ? In 20 firkins ? 39. How many whole numbers in 1? In 42 ? In 23? 40. How many whole numbers in 16 ? Ing? In 32 ? In 42 ?

41. If a skein of silk is worth 31 cents, what are 6 skeins worth?

ILLUSTRATION. If i skein is worth 34 cents, 6 skeins are worth 6 times as much; 6 times 31 are equal to 6 times 3 and 6 times ; 6 times 3 18; 6 times 1 g=3; 18 cents + 3 cents 21 cents.

Ans. 21 cents. 42. Bought one pair of boots for 64 dollars ; what must I pay for 4 pairs ? For 8 pairs ? For 10 pairs ? For 12 pairs ?

43. Paid 124 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 43 pounds worth?

ILLUSTRATION. — If 1 pound is worth 12 cents, 44 pounds are worth 44 times as much; 44 times 12 cents are equal to 4 times 12 cents and of 12 cents; 4 times 12 cents are 48 cents, and 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 9} pounds ?

46. Bought 1 pound of coffee at 16 cents; what will 53 pounds cost ? 37 pounds ? 57 pounds ? 64 pounds ?

47. If 1 yard of cloth is worth 20 cents, what is the value of 164 yards ? 124 yards ? 81 yards ? 111 yards ?

48. If 11 bushels of corn cost 120 cents, what will 1 bushel cost ?

ILLUSTRATION. — 14 bushels of bushels. Now, if cost 120 cents, will cost of 120 cents, or 40 cents; and f, 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?

1

ILLUSTRATION. -24 pounds pounds. If y cost 60 cents, } will cost t of 60 cents, or 5 cents; and, or a pound, will cost 5 times 5 cents, or 25 cents.

Ans. 25 cents. 50. How many times will 60 contain 2 ?

51. Paid 54 dollars for 7$ barrels of oil ; what cost one barrel?

Ans. 7 dollars. 52. How many times is 7 contained in 54 ? 53. How

many

cords of wood at 54 dollars per cord, can be bought for 66 dollars ?

54. How many times will 66 contain 51 ?

55. Gave forty dollars for 64 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 54 rods each. Required the number of spaces ?

CONTRACTIONS.*

CONTRACTIONS IN MULTIPLICATION.

OPERATION.

61. To multiply by 25.
Ex. 1. Multiply 876581 by 25.

Ans. 21914525.

We multiply by 100, by annex4) 8 7 6 5 8100

ing two ciphers to the multipli2 1 9 1 4 5 2 5 Product.

cand; and since 25, the multi

plier, is only one fourth of 100, we divide by 4 to obtain the true product.

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

* 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 ?

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