this new quotient-figure, and subtract the product from the partial dividend. Proceed in this manner until the whole dividend has been divided; the entire quotient is 358.

Every quotient-figure is of the same order as the right-hand figure of the dividend used in obtaining that quotient-figure; thus in Ex. 33, the 46 of the dividend is hundreds, and the 3 of the quotient is also hundreds; the 75 is tens and the 5 of quotient is also tens; etc.

78. This process, called Long Division, usually employed when the divisor is large, may be performed by the following

RULE 1. Write the divisor and dividend as in Short Division.

2. Divide the smallest number of figures in the left of the dividend that will contain the divisor, and set the result as the first figure of the quotient at the right of the dividend.

3. Multiply the divisor by the quotient-figure, and set the product under that part of the dividend taken.

4. Subtract the product from the figures over it, and to the remainder annex the next figure of the dividend for a new partial dividend.

5. Divide, and proceed as before, until the whole dividend has been divided.

NOTE 1. It will be seen that the process of dividing consists of four distinct steps, viz.: first, to seek a quotient figure; second, multiply; third, subtract; and, fourth, form a new partial dividend by annexing the next figure of the dividend to the remainder.

NOTE 2. If any partial dividend will not contain the divisor, 0 must be placed in the quotient, and another figure brought down and annexed to the dividend.

78. When is Long Division employed? Give the rule for Long Division. How many steps in dividing? What are they? Repeat Note 2.

NOTE 3. If the product of the divisor multiplied by the quotient figure is greater than the partial dividend, the quotient figure is too large, and must be diminished.

NOTE 4. If the remainder equals or exceeds the divisor, the quotient is too small, and must be increased.

79. In the same manner solve the following examples, dividing each upper number by the one under it in each example; also, in the same manner, as suggested by the signs.

80. Division is the reverse of multiplication. In multiplication, the two factors are given, and the product is required; in division the product and one factor are given, and the other factor is required. The dividend is the product, and the divisor and quotient are the factors; thus,

PROOF. Multiply the divisor by the quotient, and to the product add the remainder; the SUM should be the dividend.

78. Repeat Note 3. Note 4. 80. What is said of Division and Multipli cation? In Multiplication what is given? What required? In Division what is given? Required? How is Division proved?

50. A farm containing 327 acres, was bought for $ 37605;

what was the price per acre?

Ans. $115.

51. Divide six thousand eight hundred and forty-four acres of land into twenty-nine equal parts. Ans. 236 acres.

52. A drover paid $2331 for 37 oxen; what was the average price per ox? Ans. $63. 53. The product of two numbers is 35068765, and one of the numbers is 8765; what is the other number?

Ans. 4001.

54. In how many days will a steamboat sail 11352 miles, if she sails 264 miles per day?

55. If a railroad 359 miles long cost $3545484, what was the average cost per mile?

Ans. $9876.

CONTRACTIONS.

81. To divide by a composite number.

56. Divide $1855 equally among 35 men.

OPERATION.

35 = 7 X 5.

The 35 men may

be separated into 7 groups of 5 men each.

1st Factor, 7) $ 1 8 5 5 Dividend.

Then dividing by 7

2d Factor,

5) $2 6 5 1st Quotient.

gives $265 for each

$5 3 True Quotient, group, and dividing

the $265 by 5 gives $53 for each man.

NOTE. When a composite number is made up of different sets of factors, as in Ex. 57, it is immaterial which set is taken. It is also immaterial in what order the factors are taken.

81. Rule for dividing by a composite number? Is it material which factor of the divisor is used first, or which set of factors is employed?

57. Divide 10656 by 288.

2884 × 6 × 12 = 6 X 6 X 8 = 8 X 3 X 12, etc.

RULE. Divide the dividend by one factor of the divisor, and the quotient so obtained by another factor, and so on till all the factors of the set have been used. The last quotient will be the

63. Divide 33696 by 144; = 12 × 12.

82. In dividing by the factors of the divisor, there may be a remainder, after either or each of the divisions.

Should the learner find a difficulty in determining the true remainder, he has but to remember that it is always of the same kind as the dividend. (Art. 69, Note).

82. Rule for finding the true remainder when the factors of the divisor are used separately? The reason?