46. Divide 165'217 by 3.

Operation.

3) 1 6 5 217

55072

Quotient.

47. How many yards of cloth, at $5 a yard, can be

bought for $95 ?

Ans. 19.

Note.--There will be as many yards as there are 5's in 95.

48. What is a 6th part of 65'057 ?

Ans. 10'842§.

49. What is the eighth part of one thousand? Ans. 125.

50. A gentleman divides his estate, amounting to 11'097 dollars, equally between nine children: what does each receive? Ans. $1'233.

51. How many spaces of time, each 5 minutes, in 120 minutes? Ans. 24. 52. If a stick of timber 89 feet long, be divided into 7 equal parts, what will be the length of each part? Ans. 12 feet. 53. If a stick of timber 100 feet long, be divided into divisions of 8 feet each, how many divisions will it consist of? Ans. 12.

Contraction 2.

When any of the places at the right of the divisor, are occupied by ciphers, they may be omitted, and an equal number of figures must be omitted at the right of the dividend. The figures cut off from the dividend, must be brought to the right of the remainder, if any, if not, they will constitute the whole remainder.

54. What is the quotient of 8'990'123, divided by 76'300 ?

Operation.

76 300) 8 9 9 0 1 2 3 1 17
76300

Illustration of contraction 2.-We obtain the same quotient, whether we omit the ciphers and an equal number of figures at the right hand of the dividend, or retain them, because the ciphers on the right of the divisor, will always come under an equal number of figures brought from the right of the dividend; and as they do not lessen the figures above them by subtracting, the figures above them the last time we subtract, which were brought from the right hand of the dividend, will always be left as a remainder, or a part of it.

By examining the two last operations, it will be seen that we obtain the same quotient and the same remainder, by both methods.

The scholar is required to perform the following examples, as directed by the sign of division :

55. 4'521'084-÷3'600=1'255, and 3'084 rem. 56. 9'876'543'210-432'000=22'862, and 159'210 rem.

57. 2'468'013'579999'000=2'470, and 483'579 rem.

58. 5'489'074'426-500'000=10'978, and 74'426

rem.

59. 8'296'547'000--9'000'000 921, and 7'547'000 rem.

60. 98'899'887'701-9'001'000=10'987, and 5'900'701 rem.

Contraction 3.

When the divisor is 1 with a cipher, or ciphers, to the right hand, we have only to cut off as many figures at the right hand of the dividend, as there are ciphers in the divisor;-the figures cut off will be the remainder, and the others, the quotient. 61. Divide 123'456'789 by 10'000.

Operation by the common method.

1 0 0 0 0) 1 3 4 5 6 7 8 9 ( 1 2 3 4 5

10000

2 3 4 5 6
20000

3 4 5 6 7
30000

45678 4 0 0 0 0

56789
50000

6 7 8 9 Rem.

Illustration of contraction 3.-The figure 1 in the divisor, can be taken from every figure in the dividend as many times as there are 1's in the figures of the dividend; consequently, the figures in the quotient must be the same as the left hand figures of the dividend. As the ciphers in the divisor will come under an equal number of figures at the right hand of the dividend, those figures will always remain. See the operation and contraction of the last example.

62. What is the quotient of ninety thousand and nine, divided by one thousand. Ans. 90° 63. Divide one million and one, by one million.

Quot. 1ōō •

64. A gentleman said that he owned a tenth part of a tract of land, valued at $4'603; what is the value of his share? Ans. $460,

65. What is a 100th, 1000th, 100000th part of 745'638'965 ?

Questions to be worked by Addition and Division.

1. A farmer sells 21 barrels of cider for $41, and 19 barrels for $39; what does the cider average a barrel? Ans. $2. 2. A marketman sells 3 turkeys, weighing 9, 10, and 13 pounds, for $3,52; what does he receive per pound? Ans. 11 cents.

3. From the creation of the world to the birth of Christ, was 1'461'460 days; from the birth of Christ to January, 1826, was 666'490 days. Estimating

365 days in a year, how many years from the creation to January, 1826? Ans. 5'830.

Questions to be worked by Multiplication and Division.

1. A gentleman divided 4 shares of canal stock equally between 5 sons. Now, if each share is worth $597, what does each son receive? Ans. $4773.

2. A man buys 165 pounds of butter for 15 cents a pound. In market he sells it for 17 cents a pound, and receives for the whole $27,71; by which he finds it had lost in weight: how many pounds did he sell? and what did he give for the whole?

Ans. 163 pounds; he gave $24,75. 3. A merchant sells 3 chests of tea, each weighing 96 pounds, for $118,08; what does he receive a pound? Ans. 41 cents.

Questions to be worked by Subtraction and Division.

1. If the difference between 307'199 and 126'900, be divided into 9 equal parts, what number will one of the parts be? Ans. 20033.

2. Massachusetts contains 7'250 square miles, and 529'250 inhabitants; New-York, 46'000 square miles, and 1'610'000 inhabitants. Which State contains the greatest number of inhabitants to a square mile? Ans. Mass. contains 38 more than N. Y.

Questions to be worked as directed by the signs.

1. 10732+720-67 x 13÷110=1345,55. 2. 67276+2-6×46-8.

3. A boy told his companion, that he had apples enough to equal 18 x 2÷3÷12x3+2+5+1; how many had he? Ans. 2.

4. A gentleman being asked how many dollars his daily income was, answered, that a 10th part of 40 plus 2 into 4 by 2 minus 8 by 4, would be the number; what was his daily income? Ans. $1.