1 under the 7, and suppose 1, the remainder, to be placed before the next figure of the dividend, 5 ; and the number would be 15. Then inquire how many times 6, the divisor, is contained in 15. It is found to be 2 times, and 3 remaining. Write the 2 under the 5, and suppose the remainder, 3, to be placed before the next figure of the dividend, 5; and the number would be 35. Inquire again how many times 35 will contain the divisor, 6. It is found to be 5 times, and 5 remaining. Write the 5 under the 5 in the dividend, and suppose the remainder, 5, to be placed before the last figure of the dividend, 4; and the number would be 54. Lastly, inquire how many times 54 will contain the divisor, 6. It is found to be 9 times, which we place under the 4 in the dividend. Thus we find, that each man will receive 1259 dollars.

From the above illustration we deduce the following

RULE.

See how many times the divisor may be contained in the first figure or figures of the dividend, and place the result immediately under that figure; and what remains suppose to be placed directly before the next figure of the dividend; and then inquire how many times these two figures will contain the divisor, and place the result as before; and so proceed until the question is finished.

16. Divide 944,580 dollars equally among 12 men, and what will be the share of each ? Ans. 78,715 dollars. 17. Divide 154,503 acres of land equally among 9 perAns. 17,167 acres.

sons.

What

18. A plantation in Cuba was sold for 7,011,608 dollars, and the amount was divided among 8 persons. was paid to each person?

Ans. 876,451 dollars.

25. A prize, valued at 178,656 dollars, is to be equally divided among 12 men; what is the share of each ? Ans. 14,888 dollars. 26. Among 7 men, 67, 123 bushels of wheat are to be distributed; how many bushels does each man receive? Ans. 9,589 bushels.

8

27. If 9 square feet make 1 square yard, how many yards in 895,347 square feet? Ans. 99,483 yards. 28. A township of 876,136 acres is to be divided among persons; how many acres will be the portion of each? Ans. 109,517 acres. 29. Bought a farm for 5670 dollars, and sold it for 7896 dollars, and I divide the net gain among 6 persons; what does each receive? Ans. 371 dollars. 30. If 6 shillings make a dollar, how many dollars in 7890 shillings?

Ans. 1315.

II. When the divisor exceeds 12, the operation should be performed by

LONG DIVISION,

as in the following question.

31. A gentleman divided equally among his 19 sons, 4712 dollars; what is the share of each?

OPERATION.

Dividend.

The object of this question is to find

Divisor. 19)4712( 248 Quotient. how many times

38

19

91 2232

76 248

152

1524712 Proof.

000 Remainder.

4712 will contain 19, or how many times 19 must be subtracted from 4712, until nothing remains. We first inquire how many times 19 may be contained in 47 (thousand). Having found it to be 2 (hundred) times, we write 2 in the quotient and multiply it by the divisor, 19, and place their product under 47, from which we subtract it, and find the remainder to be 9, to which we annex the next figure in the dividend, 1. And having found that 91 (tens) will contain the divisor, 19, 4 (tens) times, we write 4 in the quotient, multiply it by 19, and place the product 76 under 91, from which we subtract it, and, to the remainder, 15 (tens), we annex the last figure of the dividend, 2, and inquire how many times 152 will contain 19, and we find it to be 8 times; and having placed the product of 8 times 19, that is, 152, under the 152, we find there is no remainder, and that the number 4712 will contain 19, the divisor, 248 times; that is, each man will receive 248 dollars.

To prove our operation is correct, we reason thus. If one man receive 248 dollars, 19 men will receive 19 times as much, and 19 times 248 are 4712, the same as the dividend; and this operation is effected by multiplying the divisor by the quotient, and adding in the remainder if there be one. The student will now see the propriety of the following

RULE.

Place the divisor before the dividend, and inquire how many times it is contained in a competent number of figures in the dividend, and place the result in the quotient; multiply the figure in the quotient by the divisor, and place the product under those figures in the dividend, in which it was inquired, how many times the divisor was contained; subtract this product from the dividend, and to the remainder

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.

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 numer ator. This is the analytical method.