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9 times. Set the 9 in the quotient, multiply and subtract, and the remainder is 2. Write the 2 over the divisor, and annex it to the quotient. (Art. 58.)

64. After the first quotient figure is obtained, for each figure of the dividend which is brought down, either a significant figure or a cipher must be put in the quotient.

Solve the following examples in a similar manner: 10. Divide 15425 by 11.

Ans. 1402 r. 11. Divide 31237 by 15.

Ans. 208215 65. From the preceding illustrations and principles we derive the following

RULE FOR LONG DIVISION. I. Beginning on the left of the dividend, find how many times the divisor is contained in the fewest figures that will contain it, and place the quotient figure on the right of the dividend with a curve line between them.

II. Multiply the divisor by this figure and subtract the product from the figures divided ; to the right of the remainder bring down the next figure of the dividend, and divide this number as before. Procced in this manner till all the figures of the dividend are divided.

III. When there is a remainder after dividing the last figure, write it over the divisor, and annex it to the quotient, as in short division.

Obs. 1. Long Division is proved in the same manner as Short Division.

2. When the divisor contains but one figure, the operation by Short Division is the most expeditious, and should therefore be practiced; but when the divisor contains two or more figures, it will generally be the most convenient to divide by Long Division.

QUEST.-65. How do you divide in long division? Where place the quotient? After obtaining the first quotient figure, how proceed? When there is & remainder after dividing the last figure of the dividend what must be done with it? Ols. How is long division proved? When should sbort division be usod? When long division ?

EXAMPLES FOR PRACTICE.

1. Divide 369 by 8. 10. Divide 675 by 25. 2. Divide 435 by 9. 11. Divide 742 by 31. 3. Divide 564 by 7. 12. Divide 798 by 37. 4. Divide 403 by 10. 13. Divide 834 by 42. 5. Divide 641 by 11. 14. Divide 960 by 48. 6. Divide 576 by 12. 15. Divide 1142 by 53. 7. Divide 274 by 13. 16. Divide 2187 by 67. 8. Divide 449 by 14. 17. Divide 3400 by 75. 9. Divide 617 by 15. 18. Divide 4826 by 84.

19. How many caps, at 7 shillings apiece, can I buy for 168 shillings?

20. How many pair of boots, at 5 dollars a pair, can be bought for 175 dollars ?

21. A man laid out 252 dollars in beef, at 9 dollars a barrel : how many barrels did he buy?

22. In 12 pence there is 1 shilling: how many shillings are there in 198 pence?

23. In 20 shillings there is 1 pound : how many pounds are there in 215 shillings?

24. In 16 ounces there is 1 pound : how many pounds are there in 268 ounces?

25. How many trunks, at 15 shillings apiece, can be bought for 255 shillings?

26. If 27 pounds of flour will last a family a week, how long will 810 pounds last them?

27. How many yards of broadcloth, at 23 shillings per yard, can be bought for 756 shillings ?

28. If it takes 18 yards of silk to make a dress, how many dresses can be made from 1350 yards ?

29. How many sheep, at 19 shillings per head, can be bought for 1539 shillings?

30. A farmer having 1840 dollars, laid it out in land, at 25 dollars per acre: how many acres did he buy?

31. A man wishes to invest 2562 dollars in Railroad stock : how many shares can he buy, at 42 dollars per share ?

32. In 1 year there are 52 weeks: how many years are there in 1640 weeks ?

33. In one hogshead there are 63 gallons: how many hogsheads are there in 3065 gallons ?

34. If a man can earn 75 dollars in a month, hc w many months will it take him to earn 3280 dollars ? 35. If a young man's

expenses are 83 dollars a month, how long will 4265 dollars support him?

36. A man bought a drove of 95 horses for 4750 dollars : how much did he give apiece?

37. If a man should spend 16 dollars a month, how long will it take him to spend 172 dollars ?

38. A garrison having 2790 pounds of meat, wished to have it last them 31 days: how many pounds could they eat per day?

39. How many times is 54 contained in 3241, and how many over ?

40. How many times is 68 contained in 7230, and how many over ? 41. How

many

times is 39 contained in 1042, and how many over ? 42. How

many

times is 47 contained in 2002, and how many over?

43. What is the quotient of 1704 divided by 56 ? 44. What is the quotient of 2040 divided by 60 ? 45. What is the quotient of 2600 divided by 49 ? 46. What is the quotient of 2847 divided by 81 ? 47. Divide 1926 by 75. 51. Divide 9423 by 105. 48. Divide 2230 by 85. 52. Divide 13263 by 112. 49. Divide 6243 by 96. 53. Divide 26850 by 123, 50. Divide 8461 by 99. 54. Divide 48451 by 224,

66. It bas been shown that annexing a cipher to a number, increases its value ten times, or multiplies it by 10. (Art. 44.) Reversing this process, that is, removing a cipher from the right hand of a number, will evidently diminish its value ten times, or divide it by 10; for, each. Sigure in the number is thus restored to its original place, and consequently to its original value. Thus, annexing a cipher to 12, it becomes 120, which is the same as 12 X 10. On the other hand, removing the cipher from 120, it becomes 12, which is the same as 120-10.

In the same manner it may be shown, that removing two ciphers from the right of a number, divides it by 100; removing three, divides it by 1000; - removing four, divides it by 10000, &c. Hence,

67. To divide by 10, 100, 1000, &c.

Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor. The remaining figures of the dividend will be the quotient, and those cut off the remainder.

55. Divide 2456 by 109.

Since there are 2 ciphers on Operation. the right of the divisor, we cut 1100)2456 off 2 figures on the right of the Quot. 24 and 56 rem dividend. The quotient is 24 and 56 remainder, or 24 PO.

50. Divide 1325 by 10. Ans. 132 and 5 rem.
57. Divide 4626 by 100.
58. Divide 5633 by 1000.
59. Divide 8465 by 1000.
60. Divide 26244 by 1000.
61. Divide 136056 by 10000.

QUEST.-60. What is the effect of annexing a cipher to a number? What is the effect of removing a cipher from the right of a ne inber? 67. How proceed when the divisor is 10, 100, 1000, &c.?

62. Divide 2443667 by 100000. 63. Divide 23454631 by 1000000.

68. When there are ciphers on the right hand of the divisor.

Cut off the ciphers from the divisor ; also cut off as many figures from the right of the dividend. Then divide the remaining figures of the dividend by the remaining fig. ures of the divisor, and the result will be the quotient.

Finally, annex the figures cut off from the dividend to the remainder, and the number thus formed will be the true remainder.

64. At 200 dollars apiece, how many carriages can be bought for 4765 dollars ?

Having cut off the two ciphers on Operation. the right of the divisor, and two fig- 2|00)47 65 ures on the right of the dividend, we Ans. 23-165 rem. divide the 47 by 2 in the usual way. 65. Divide 2658 by 20.

Ans. 132 and 18 rem., or 13214. 66. 3642 by 30.

67. 6493 by 200 68. 76235 by 1400. 69. 82634 by 1600. 70. 93600 by 2000. 71. 14245 by 3000. 72. 23148 by 1200. 73. 42061 by 1500. 74. 50382 by 1800. 75. 88317 by 2100. 76. 894000 by 2500. 77. 9203010 by 3100. 78. 7450000 by 420000. 79. 9000000 by 300000. 80. 348676=235.

81. 467342:341. 82. 762005--401.

83. 506725-603. 84. 607507: 1623.

85. 736241-2764. 86. 43672382367.

87. 6203451-3827. 88. 8230732=-3478.

89. 8235762542316. 90. 93670858--67213. 91. 98765421-84327.

Qurst.--18. When there are ciphers on the right of the divisor, how pro ceed? What is to be done with figures cut off from the dividend ?

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