4. Divide 8040 by 8.

1005

In this example, we say 8 in 8, 1 time, 8)8040 and set 1 in the quotient. We then say, 8 in 0, 0 times, and set the 0 in the quotient: then say, 8 in 4, 0 times, and set the 0 in the quotient: then say 8 in 40, 5 times, and set the 5 in the quotient. The true quotient is therefore 1005.

§ 43. It may be remarked that any number contains 1 as many times as there are units in the number, or that if any number be divided by 1, the quotient will be equal to the number itself.

CASE I.

§ 44. When the divisor does not exceed 12.

RULE.

I. Set down the divisor on the left of the dividend, draw a curved line between them, and a straight line. under the dividend.

II. Find how often the divisor is contained in the left hand figure or figures of the dividend, and place the figure so found under the straight line, for the first figure of the quotient.

III. If there is no remainder, divide the next figure of the dividend for the next figure of the quotient. But when there is a remainder consider it as tens, to which add the next figure of the dividend, regarded as units, and divide this sum, for the next figure of the quotient, and do the same for each of the figures of the dividend.

IV. When any of the figures, or sums, that are to be divided, is less than the divisor, set down 0 in the quotient, and to such figures regarded as tens, add the next figure of the dividend considered as unit and divide the sum for the next figure of the quotie

(4)

5) 36458

7291-3 rem.

In this example we find the quotient to be 7291 and a remainder 3. This 3 ought in fact to be divided by the divisor 5; but the division cannot be effected, since 3 does not contain 5. The division must then be expressed by placing 5 under the 3, thus, 3. The true quo tient, therefore, is 72913, which is read, seven thousand two hundred and ninety one, and three divided by five.

45. Therefore, when there is a remainder after division, it must be written after the quotient, and the divisor placed under it.

5. Divide 6794108 by 3.

6. Divide 21090431 by 9.

7. Divide 2345678964 by 6.

8. Divide 570196382 by 12.

Ans. 22647023.

Ans. 23433814.

Ans. 390946494.

Ans. 47516365

NOTE. This method of dividing one number by another is called Short Division.

CASE II.

46. When the divisor contains several figures.

RULE.

I. Set down the divisor on the left of the dividend, draw a curved line between them, and also a curved line on the right of the dividend.

II. Note the fewest figures of the dividend, counted from the left hand, that will contain the divisor ; find how often they contain it, and set the figure in the quotient.

III. Multiply the whole divisor by this figure; set the product under the first figures of the dividend, and subtract it from them. To the remainder annex the next figure of the dividend, then find how often the divisor is contained in this new number, and set the figure in the quotient.

IV. Multiply the whole divisor by the last figure of the quotient, and subtract the product from the last number containing the divisor. To the remainder annex the next figure of the dividend, and find the figures of the quotient in the same way, till all the figures of the dividend are brought down.

NOTE 1. When any one of the products is greater than the number supposed to contain the divisor, the quotient figure is too large, and must be diminished.

2. When any one of the remainders is greater than the divisor, the quotient figure is too small, and must be increased by at least 1.

3. If after having annexed the figure from the dividend, to any one of the remainders, the number is less than the divisor, the quotient figure is 0, which being written in the quo. tient, annex the next figure of the dividend and divide as fore.

be.

Ex. 1. Divide 11772 by 327.

Having set down the divisor on

Divisor.

Dividend.

Quotient.

981

1962

1962

the left of the dividend, it is seen 327)11772(36 that 327 is not contained in 117; and by observing that 3 is contained in 11, 3 times and something over, we conclude that the divisor is contained at least 3 times in the first four figures of the dividend.

0000

Setting down the 3 in the quotient, and multiplying the divisor by it, we get 981, which being less than 1177, the quotient figure is not too great: we subtract 981 from the first four figures of the dividend, and find a remainder 196, which being less than the divisor, the quotient figure is not too small. Annex to this remainder the next figure 2, of the dividend.

As 3 is contained in 19, 6 times, we conclude that the divisor is contained in 1962 as many as 6 times. Setting down 6 in the quotient and multiplying the divisor by it, we find the product to be 1962. Therefore the entire quotient is 36, or the divisor is contained 36 times in the dividend.

DEMONSTRATION OF THE RULE.

If 6 simple units be divided by 3, the quotient will be 2. If 6 units of the 2d order, or 60, be divided by 3, the quotient will be 2 tens, or 2 units of the 2d order. If 9 hundreds, or 9 units of the 3d order be divided by 3, the quotient will be 3 hundreds, or 3 units of the 3d order.

So, in general, if units of any order be divided by simple units, the units of the quotient will be of the same order as those of the dividend.

327)11772(36

981

1962
1962

When in the last example it was required to divide 11772 by 327, we first considered, as we had a right to do, that 11772 is made up of 1177 tens and 2 units. We then divided the tens by the divisor 327, and found 3 tens for the quotient, by which we multiplied the divisor and subtracted the product from 1177, leaving a remainder of 196 tens. To this number we brought down the 2 units, making 1962 units. This numer contained the divisor 6 times: that is, 6 unit's

nes.

When the unit of the first number which contains the divisor is of the 3d order, or 100, there will be 3 figures in the quotient; when it is of the 4th order there will be 4, &c.

Hence, the quotient found according to the rule, expresses the number of times which the dividend contains the divisor, and consequently is the true quotient.

PROOF OF DIVISION.

47. Multiply the divisor by the quotient and add in the remainder, when there is one: the sum should be equal to the dividend.

Ex. 1. Divide 67289 by 261.

In this example we find a quotient of 257 and a remainder of 212, which being less than the divisor will not contain it.

261)67289(257

522

1508

1305

2039