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bring down the next figure of the dividend ; and then inquire how many times this number will contain the divisor, and place the result in the quotient, and proceed as before, until all the figures of the dividend are brought down.

Note 1.- It will sometimes happen, that, after a figure is brought down, the number will not contain the divisor; a cipher is then placed in the quotient, and another figure is brought down, and so continue until it will contain the divisor, placing a cipher each time in the quotient.

NOTE 2. — The remainder in all cases is less than the divisor, and of the same denomination of the dividend; and, if at any time, we subtract the product of the figure in the quotient and divisor from the dividend, and the remainder is more than the divisor, the figure in the quotient is not large enough.

PROOF.

Division may be proved by Multiplication, Addition, or by Division itself.

To prove it by Multiplication, the divisor must be multiplied by the quotient, and, to the product, the remainder must be added, and, if the result be like the dividend, the work is right.

To prove it by Addition. Add up the several products of the divisor and quotient with the remainder, and, if the result be like the dividend, the work is right.

To prove it by Division itself. Subtract the remainder from the dividend, and divide this number by the quotient, and the quotient found by this division will be equal to the former divisor, when the work is right. 32.

33. 83)1486780 1791 427)5 6 7 896(1329 83*

83

427
6 5 6
5373

1408
581 *
14328

1281
757 148653

1279
74 7*
25

854
108 1486 78 Proof. 4 2 56
8 3*

3843

413 148678 * Note. The asterisms show the numbers to

be added.

2 5*

35.

34.
144) 13 8 24 (96 96) 138 24(144
1296

96
864

422 864

384 NOTE. The 34th question is proved by

384 the 35th.

384

87.

55.

36.

37.

Rem. 861000) 8963 | 486 ( 104 1|0000 ) 7|8967 ( 7 Quotient.

86
363
344
19486

Quotients. Remainders. 38. Divide 867532 by

59. 14703. 55. 39. Divide 167008 by

1919. 40. Divide 345678 by 379.

912. 30

3037 41. Divide 6789563 by 1234.

95. 42. Divide 78112345 by 8007.

4060. 43. Divide 34533669 by 9999.

7122. 44. Divide 99999999 by 3333.

0. 45. Divide 47856712 by 1789.

962. 46. Divide 13112297 by 8900.

2597. 47. Divide 10000000 by 7007.

1011. 48. Divide 15678953 by 8790.

6383. 49. Divide 71800100 by 4701.

1727. III. To multiply by a fraction.

RULE.

Multiply the given number by the numerator of the fraction, and divide the product by the denominator. If any thing remain place it over the divisor at the right hand of the quotient.

Note. When the number is such, that it may be divided by the denominator without a remainder, the better way is to divide the given number by the denominator, and multiply the quotient by the numerator. This is the analytical method.

one

50. What is of 144 ?
Synthetic method.

Analytical method.
144
4)144

Divide by 4 to 3

36

get fourth, 4) 432

3

and multiply by 3 10 8 Ans.

108 Ans.

to get 3 fourths. 51. What is of 365 ? Ans. 228). 52. What is of 128 ? Ans. 546. 53. What is A of 386 ? Ans. 210 41. 54. Sold a farm for 1728 dollars ; and, if I give a of this sum to indigent persons, what do they receive ?

Ans. 720 dollars. 55. If from 1000 dollars f be taken, what sum will remain ?

Ans. 625 dollars. IV. To divide by a fraction.

RULE.

Multiply

the given-number by the denominator, and divide the product by the numerator. 56. Divide 125 by 125

In this example, we mul8

tiply by 8 to reduce the 125 5)1000 to eighths ; and then we 200

see how often 5 (eighths)

are contained in them. 57. Sold of a house for 3227 dollars ; what was the value of the whole house ?

Ans. 3688 dollars. V. To divide by a composite number, that is, a number, which is produced by the multiplying of two or more numbers.

RULE.

Divide the dividend by one of these numbers, and the quotient thence arising by another, and so continue ; and the last quotient will be the answer.

Note. To find the true remainder, we multiply the last remainder by the last divisor but one, and to the product add the next preced

OPERATION.

ing remainder; we multiply this product by the next preceding divisor, and to the product add the next preceding remainder; and so on until we have gone through all the divisors and remainders to the first. 58. Divide 67872 by 24.

In this question, we divide 4)67872 by 4 and 6, because they are 6) 16968

the factors, or composite num

bers of 24. 2828

Quotients. 59. Divide 765325 by 25=5 X 5.

30613. 60. Divide 123396 by 84=7 x 12.

1469. 61. Divide 611226 by 81=9x9.

7546. 62: Divide 987625 by 125=5X5 X 5. 7901.

Section 6.

APPLICATION OF THE PRECEDING RULES. 1. A farmer bought 5 yoke of oxen at 87 dollars a yoke ; 37 cows at 37 dollars each ; 89 sheep at 3 dollars a piece. He sold the oxen at 98 dollars a yoke ; for the cows he received 40 dollars each ; and, for the sheep, he had 4 dollars a piece ; what did he gain by his trade ?

Ans. 255 dollars. 2. In 4008 hours, how many days? Ans. 167 days. 3. In 169 weeks, how many days ? Ans. 1183 days. 4. If 12 inches make a foot, how many feet in 48096 inches ?

Ans. 4008 feet. 5. In 15300 minutes, how many hours ?

Ans. 255 hours. 6. If 144 inches make 1 square foot, how many square feet in 20736 inches ?

Ans. 144 feet. 17. An acre contains 160 square rods ; how many in a farm containing 769 acres ? Ans. 123040 rods. 8. A gentleman bought a house for three thousand fortyseven dollars, and a carriage and span of horses for five hundred seven dollars. He paid at one time, two thousand seventeen dollars, and at another time, nine hundred seven dollars. How much remains due ?

Ans. 630 dollars.

9. The erection of a factory cost 68,255 dollars ; suppos

ing this sum to be divided into 365 shares, what is the expense of each ?

Ans. 187 dollars. 10. A gentleman, possessing an estate of 375,846 dollars, bequeathed 7,494 dollars to the Bible Society ; 4,230 dollars for the support of schools ; and one third to his wife ; the remainder was to be equally divided among his 12 sons and 8 daughters; what sum will each receive ?

Ans. 11,942 dollars. 11. There were distilled in the United States in 1840, thirty-six millions three hundred forty-three thousand two hundred thirty-six gallons of ardent spirits ; and the number of free white males, over 15 years, is four millions seventy-four thousand nine hundred fifteen ; now supposing the liquor to be drank by one third of those persons in one year, what quantity would each consume?

Ans. More than 26 gallons. 12. A man gave half of his estate to his wife, one third of what remained to his son, and the residue was equally divided among his 7 daughters, who received each 124 dollars.; what was the whole estate ?

Ans. 2,604 dollars.

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