34. Daniel has 47 apples; he gave William 7 and Samuel 10; how many will he have, if he share the remainder with himself and 2 brothers?

Let the pupil perform the last 16 questions on the slate, and hence notice, that

DIVISION is a short or compendious way of performing Subtraction.

Its object is to find how many times one number is contained in another. It consists of four parts; the dividend or number to be divided; the divisor or the number to be divided by; the quotient, which shows how many times the divisor is contained in the dividend; and the remainder, which is always less than the divisor, and of the same name of the dividend.

When the divisor is less than 13, the question should be performed by

SHORT DIVISION.

EXAMPLE.

1. Divide 948 dollars equally among 4 men.

Dividend.

Divisor 4)948
Quotient 237

In performing this question, inquire how many times 4, the divisor, is contained in 9, which is 2 times, and 1 remaining; write the 2 under the 9 and suppose 1, the remainder, to be placed before the next figure of the dividend, 4, and the number would be 14. Then inquire how many times 4, the divisor, is contained in 14. It is found to be 3 times and 2 remaining. Write the 3 under the 4, and suppose the remainder, 2, to be placed before the next figure of the dividend, 8, and the number would be 28. Inquire again how many times 28 will contain the divisor. It is found to be 7 times, which we place under the 8. Thus we find each man receives 237 dollars. From the above illustration, we deduce the following

RULE.

See how many times the divisor may be had 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; ana so proceed until the question is finished.

When the divisor exceeds 12, the operation is performed by

as in the following

LONG DIVISION,

EXAMPLE.

1. A prize, valued at $ 3978, is to be equally divided among 17 men. What is the share of each?

OPERATION.

Dividend.

The object of this question is to find, how many

Divisor. 17) 3978 ( 234 Quotient. times $978 will contain 17,

34

17

57

1638

51

234

68 3978 Proof.
68

00 Remainder.

or how many times must 17 be subtracted from 3978, until nothing shall remain. We first inquire, how many times the first two figures of the dividend will contain the divisor; that is, how many times 39 (thousand) will contain 17. Having found it to be 2 times, we write 2 in the quotient and multiply it by the divisor, 17, and place their product 34 under 39, from which we subtract it, and find the remainder to be 5, to which we annex the next figure of the dividend, 7. And having found that 57 will contain the divisor 3 times, we write 3 in the quotient, multiply it by 17, and place the product 51 under 57, from which we subtract it, and to the remainder, 6, we annex the next figure of the dividend, 8, and inquire how many times 68 will contain the divisor, and find it to be 4 times. And having placed the product of 4 times 17 under 68, we find there is no remainder, and that 3978 will contain 17, the divisor, 234 times; that is, each man will receive 234 dollars. To prove our work is right, we reason thus. If one man receives 234 dollars, 17 men will receive 17 times as much, and 17 times 234 are 3978, the same as the dividend; and

this operation is effected by multiplying the divisor by the quotient. 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 in 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 to be placed in the quotient, and another figure is to be 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 the diviser 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, casting out the 9's, 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 See Example 1.

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. See Example 2.

To prove it by casting out the 9's-Cast the 9's out of the divisor, and place the remainder at the teft hand of a cross; then cast them out of the quotient, and place the remainder at the right hand of the cross, and lastly subtract the remainder from the dividend, and cast the 9's out of what may remain, and place the result at the top of the cross; and if it be like the product of the figures at the sides of the cross (after the 9's are cast out of their products) the work is right. See Example 3. 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. See Example 4.

CASE II.

To divide by any number with ciphers annexed.

Cut off the ciphers from the divisor, and the same number of digits from the right hand of the dividend; then divide the remaining figures as in the last case, and the quotient is the answer; and what remains written before the figures cut off is the true remainder.

To divide by an unit with ciphers annexed.

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

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

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