37. Long Division, or 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.

IV. 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.

37. How do you set down the numbers for division? What do you do next? What do you do next? What is the next step? How many operations are there in division? Name them.

V. 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. There are five operations in division. First, to write down the numbers; second, find how many times; third, multiply; fourth, subtract; fifth, bring down.

EXAMPLES ILLUSTRATING PRINCIPLES.

OPERATION.

Divisor.

Dividend.

Quotient,

38. Let it be required to divide 11772 by 327. Having set down the divisor on the left of the dividend, it is seen that 327 is, not contained in 117; but 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 firstfour figures of the dividend.

Set down the 3 in the quotient, and multiply the divisor by it; we thus get 981, which being less than

327)11772(36

981

1962

'1962

0000

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 divi. dend.

As 3 is contained in 19, 6 times, we conclude that the

38. If any one of the products is too large, what do you do? If any one of the remainders is greater than the divisor, what do you do? What will be the order of units expressed by any figure of the quotient? When the divisor is contained in simple units, what units will the quotient figure express? When the divisor is contained in tens, what units will the quotient figure express? When it is contained in hundreds? In thousands?

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.

NOTE I. After multiplying by the quotient figure, if 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.

NOTE 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.

NOTE 3. The lowest order of units in the first dividend 1177, being tens, the first figure of the quotient will be tens; and generally,

Any figure of the quotient will express units of the same order as that expressed by the right-hand figure of its corresponding dividend.

1

OPERATION.

26)2756(106

26

39. Let it be required to divide 2756 by 26. We first say, 26 in 27 once, and place 1' in the quotient. Multiplying by 1, subtracting, and bringing down the 5, we say, 26 in 15, 0 times, and place the 0 in the quotient. Bringing down the 6, we find that the divisor is contained in 156, 6 times.

156

156

NOTE. 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 quotient, annex the next figure of the dividend and divide as before.

PROOF OF DIVISION.

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

39. When any dividend is less than the divisor, what is the quotient figure? How do you then proceed?

40. How do you prove division?

DEMONSTRATION OF THE RULE OF DIVISION.

41. Let us suppose, as an example, that it were required to divide 11772 by 327.

OPERATION.

327)11772(36

981.

1962

1962

We first consider, as we have a right to do, that 11772 is made up of 1177 tens and 2 units. We then divide the tens by the divisor 327, and find 3 tens for the quotient, by which we multiply the divisor and subtract the product from 1177, leaving a remainder of 196 tens. To this number we bring down the 2 units, making 1962 units. This number contains, the divisor 6 times; that is, 6 units' times.

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,

EXAMPLES ILLUSTRATING PRINCIPLES.

1. Let it be required to divide 67289 by 261. In this example, we find a quo

tient of 257 and a remainder of 212, which being less than the divisor will not contain it.

PROOF. 261 divisor.

257 quotient.

1827

1305

522

OPERATION.

261)67289(257 522

1508

1305

2039

1827

212 rem.

212 remainder.

67289 the dividend: hence, the work is right.

2. Let it be required to divide 119836687 by 39407.

42. When two numbers are multiplied together the multiplicand and multiplier are both factors of the product; and if the product be divided by one of the factors, the quotient will be the other factor. Hence,

If the product of two numbers be divided by the multiplicand, the quotient will be the multiplier; or, if it be divided by the multiplier, the quotient will be the multiplicand.

2. The multiplicand is 61835720, and the product 8162315040: what is the multiplier ?.

Ans. 132.

3. The multiplier is 270000: now if the product be 1315170000000, what will be the multiplicand?

4. The product is 68959488, the multiplier 96: what is the multiplicand? Ans. 718328.

42. If two numbers are multiplied together, what are the factors of the product? If the product be divided by one of the factors, what will the quotient be? How do you prove multiplication?