48. When the multiplier and multiplicand both have ciphers on the right.

Multiply the significant figures of the multiplicand by the significant figures of the multiplier, and to this product annex as many ciphers, as are found on the right of both factors.

75. Multiply 16000 by 3200.

Having placed the significant figures of the multiplier under those of the multiplicand, we multiply by them as usual, and to the product thus obtained, annex five ciphers, because there are five ciphers on the right of both factors.

Solve the following examples:

76. 2100X200. 78. 12000 X 210. 80. 38000 X19000. 82. 2800000 X 26000. 84. 1000 miles X 140. 86. 120 dollars X4200. 88. 867 pounds X424. 90. 6726 rods X 627. 92. 25268 pence X 4005. 94. 376245×3164. 96. 600400X7034. 98. 432467 × 30005. 100. 680539 X 80406.

Operation.

16000 3200

32

48

Ans. 51200000

77. 3400X130.

79. 25000X2600. 81. 500000 X 42000. 83. 140 yards X16000. 85. 20 dollars X 35000. 87. 75000 dolls. X 365. 89. 6830 feet X 562. 91. 7207 galls. × 807. 93. 36074 tons X 4060. 95. 703268×5346. 97. 864325×6728.

99. 4567832×27324. 101. 7563057 × 62043.

102. Multiply seventy-three thousand and seven by twenty thousand and seven hundred.

103. Multiply six hundred thousand, two hundred and three by seventy thousand and seventeen.

QUEST.-48. When there are ciphers on the right of both the multiplier and multiplicand, how proceed?

SECTION V.

DIVISION.

ART. 49. Ex. 1. How many lead pencils, at 2 cents apiece, can I buy for 10 cents?

Solution. Since 2 cents will buy 1 pencil, 10 cents will buy as many pencils, as 2 cents are contained times in 10 cents; and 2 cents are contained in 10 cents, 5 times. I can therefore buy 5 pencils.

2. A father bought 12 pears, which he divided equally among his 3 children: how many pears did each receive?

Solution.-Reasoning in a similar manner as above, it is plain that each child will receive 1 pear, as often as 3 is contained in 12; that is, each must receive as many pears, as 3 is contained times in 12. Now 3 is contained in 12, 4 times. Each child therefore received 4 pears.

OBS. The object of the first example is to find how many times one given number is contained in another. The object of the second is to divide a given number into several equal parts, and to ascertain the value of these parts. The operation by which they are solved is precisely the same, and is called Division. Hence,

50. DIVISION is the process of finding how many

one given number is contained in another.

times

The number to be divided, is called the dividend. The number by which we divide, is called the divisor. The answer, or number obtained by division, is called the quotient, and shows how many times the divisor is contained in the dividend.

QUEST.-50. What is division? The number by which we divide? .ne quotient show?

What is the number to be divided, called?
What is the answer called? What does

Note.-The term quotient is derived from the Latin word quoties which signifies how often, or how many times.

51. The number which is sometimes left after division, is called the remainder. Thus, when we say 4 is contained in 21, 5 times and 1 over, 4 is the divisor, 21 the dividend, 5 the quotient, and 1 the remainder.

OBS. 1. The remainder is always less than the divisor; for if it were equal to, or greater than the divisor, the divisor could be contained once more in the dividend.

2. The remainder is also of the same denomination as the dividend; for it is a part of it.

52. Sign of Division (÷). The sign of Division is a horizontal line between two dots (÷), and shows that the number before it, is to be divided by the number after it. Thus, the expression 24-6, signifies that 24 is to be divided by 6.

Division is also expressed by placing the divisor under the dividend with a short line between them. Thus the expression, shows that 35 is to be divided by 7, and is equivalent to 35÷7.

53. It will be perceived that division is similar in principle to subtraction, and may be performed by it. For instance, to find how many times 3 is contained in 12, subtract 3 (the divisor) continually from 12 (the dividend) until the latter is exhausted; then counting these repeated subtractions, we shall have the true quotient. Thus, 3 from 12 leaves 9; 3 from 9 leaves 6; 3 from 6 leaves 3; 3 from 3 leaves 0. Now, by counting, we find that 3 has

QUEST.-51. What is the number called which is sometimes left after division? When we say 4 is in 21, 5 times and I over, what is the 4 cailed? The 21 The 5? The 1? Obs. Is the remainder greater or less than the divisor? Why? Of what denomination is it? Why? 52. What is the sign of divi sion? What does it show? In what other way is division expressed?

been taken from 12, 4 times; consequently 3 is contained 12, 4 times.

Hence,

Division is sometimes defined to be a short way of performing repeated subtractions of the same number.

OBS. 1. It will also be observed that division is the reverse of multiplication. Multiplication is the repeated addition of the same number; division is the repeated subtraction of the same number. The product of the one answers to the dividend of the other: but the latter is always given, while the former is required.

2. When the dividend denotes things of one kind, or denomination only, the operation is called Simple Division.

QUEST.-Obs. When the dividend denotes things of ene denomination only,

what is the operation called?

SHORT DIVISION.

ART. 54. Ex. 1. How many yards of cloth, at 2 dol. lars per yard, can I buy for 246 dollars?

Analysis. Since 2 dollars will buy 1 yard, 246 dollars will buy as many yards, as 2 dollars are contained times in 246 dollars.

Operation.

Divisor. Dividend.

2) 246 Quot. 123 yds.

dividend denotes

Directions.-Write the divisor on the left of the dividend with a curve line between them; then, beginning at the left hand, proceed thus: 2 is contained in 2, once. As the 2 in the hundreds, the 1 must be a hundred; we therefore write it in hundreds' place under the figure divided. 2 is contained in 4, 2 times; and since the 4 denotes tens, the 2 must also be tens, and must be written in tens' place. 2 is in 6, 3 times. The 6 is units; hence the 3 must be units, and we write it in units' place. The answer is 123 yards.

Solve the following examples in a similar manner:

55. When the divisor is not contained in the first figure of the dividend, we must find how many times it is contained in the first two figures.

10. At 2 dollars a bushel, how much wheat can be bought for 124 dollars?

Since the divisor 2, is not contained in the first figure of the dividend, we find how many times it is contained in the first two figures. Thus 2 is in 12, 6 times; set

Operation. 2)124

Ans. 62 bu

the 6 under the 2. Next, 2 is in 4, 2 times. The answer is 62 bushels.