The object of this question is to find how many times

4712 will contain 19, or how many times 19 must be subtracted from 4712, un

til nothing remains. We first inquire how many times 19 Having found it to

may be contained in 47 (thousand). 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.

86 000) 8963 | 486 (104 10000) 78967 (7 Quotient.

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.

Ans. 2106.

Divide by 4 to get one fourth, and multiply by 3 to get 3 fourths.

54. Sold a farm for 1728 dollars; and, if I give of this sum to indigent persons, what do they receive ?

Ans. 720 dollars. 55. If from 1000 dollars 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.

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

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.

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 4009 hours, how many days? 3. In 169 weeks, how many days? 4. If 12 inches make a foot, how inches ?

Ans. 167 days. Ans. 1183 days. many feet in 48096 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. 7. 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.