65. What will an ox weighing 569 pounds amount to at 8 cents a pound? Ans. 4552 cts. 66. If a barrel of cider can be bought for 93 cents, what will 75 barrels cost? Ans. 6975 cts. 67. If in a certain factory 786 yards of cloth are made in one day, how many will be made in 313 days? Ans. 246018 yds. 68. A certain house contains 87 windows, and each window has 32 squares of glass; how many squares are there in the whole house? Ans. 2784 squares. 69. There are 407 wagons each loaded with 30009 pounds of coal; how many pounds are there in the whole ?

Ans. 12213663 pounds. 70. Multiply three hundred and seventy-five millions two hundred and ninety-six thousand three hundred and twenty-one, by seventy-nine thousand and twenty-four.

Ans. 29657416470704.

71. What would be the cost of 687 fothers of lead at 73 dollars a fother? Ans. $50151.

SECTION V.

DIVISION.

THE object of Division is to find how many times one number is contained in another.

Division consists of three principal parts; the Dividend, or number to be divided; the Divisor, or number by which we divide; and the Quotient, which shows how many times the dividend contains the divisor.

When the dividend contains the divisor an exact number of times, the quotient is expressed by a whole number. But when this is not the case, there will be a remainder, when the division has reached its limit, and this remainder placed above the divisor, with a horizontal line between them, will form a fraction, and should be written at the right hand of the quotient, and will be a part of it. See Example 2d, and note.

1. The Remainder may be considered a fourth term in Division, and it will always be of the same denomination with the dividend.

For the sake of convenience, Division has been divided into two kinds, Long and Short.

2. All questions in which the divisor is not more than 12 may be conveniently performed by Short Division; all others are better performed by Long Division.

SHORT DIVISION.

EXAMPLE.

1. Divide 948 dollars equally among 4 men.

In performing this question, inquire how Dividend. Divisor 4)948+ many times 4, the divisor, is contained in 9, which is 2 times, and 1 remaining; write Quotient 237 the 2 under the 9 and suppose 1, the remainder, to be placed before the next figure of the dividend, 4, and the number will 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 will 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.

Write down the dividend and place the divisor on the left, with a curved or perpendicular line drawn between them. Draw also a horizontal line under the dividend, then observe how many times the divisor is contained in the first figure or figures of the dividend (beginning at the left hand), and place the quotient figure directly under the right-hand figure of the part of the dividend that was taken. If there be no remainder, proceed to inquire how many times the divisor is contained in the next figure* of the dividend, and set down the result at the right hand of the quotient figure already obtained, or directly under the figure of the dividend, and continue the work in this manner until the whole dividend is divided. But if there be a remainder either in the first or any subsequent division, imagine the number denoting it to be placed directly before the next figure of the dividend, and ascertain the number of times the divisor is contained in the number thus formed, and place the

* If this figure be smaller than the divisor, it cannot contain it, and the figure to be placed in the quotient will be a cipher. Sometimes, as when we divide by 11 or 12, we may have two successive ciphers in the quotient, as when the divisor is 12 and the next two figures are 1's or 1 and 0. We are then obliged to proceed to a third figure in the dividend, before we can effect a proper division.

quotient figure underneath, as before. Proceed in this way until every part of the dividend is thus divided, and the result will be the quotient sought.

*From this and subsequent examples it will be seen that fractions arise from division, and are parts of a unit; that the denominator of the fraction represents the divisor, and shows into how many parts the given number or quantity is divided, and the numerator, being the remainder, shows how many units of the given quantity or dividend remain undivided. By writing the numerator over the denominator in the form of a fraction, we signify that it is to be divided by the denominator; and when placed at the right hand of the whole number in the quotient, the fraction becomes a part of the quotient, and, as such, is as much less than a unit, as the numerator is less than the denominator.

LONG DIVISION.

CASE I.

EXAMPLE.

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

17 men.

OPERATION.

Dividend.

The object of this question is to find how many times 3978 will contain

Divisor. 17) 3978 (234 Quotient. 17, or how many times

34

17

57 1638

51 234

68 3978 Proof.

68

00 Remainder.

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 will contain 17. Having found it to be 2 times, we write 2 in the quotient and multiply the divisor, 17, by it, 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 and dividend as under the preceding rule, and draw a curved or perpendicular line on the right of the dividend. Then observe how many figures of the dividend, counting from left to right, must be taken to contain the divisor one or more times, but never exceeding nine times, and ascertain how many times these figures will

contain the divisor, placing the quotient figure on the right hand of the dividend. Then multiply the divisor by this quotient figure, and place the product in order under the figures of the dividend that were taken. Subtract this product from the part of the dividend above it, and to the difference bring down and annex the next figure of the dividend. Divide this number by the divisor, and place the quotient figure on the right of the one already found. Multiply the divisor by the quotient figure last found, and subtract the product from the number last divided, and bring down and annex as before, till the last figure of the dividend is taken; and the several figures on the right of the dividend will be the quotient required. The difference between the number last divided and the last product will be the remainder, which, with the divisor, will form a fraction, as under the preceding rule.

NOTE 1. - It will often happen, that, after a figure is brought down and annexed to a remainder, the number will not contain a divisor. In such a case, a cipher is to be placed in the quotient, and the next figure to be brought down and annexed, and thus till the number formed shall be large enough to contain the divisor. Sometimes it will be necessary thus to place several ciphers in succession in the quotient.

NOTE 2. - The proper remainder is in all cases less than the divisor; and if, at any time, the subtraction named in the rule gives a remainder larger than the divisor, we discover at once, that an error has been_committed in the division, and that the quotient figure must be increased.

PROOF.

Division may be proved by Multiplication, by Addition, by casting out the 9's, or by Division.

By the first method, we multiply the quotient by the divisor, adding to the product the remainder, and the result, if the work be right, is equal to the dividend.

By the second method, we add up the several products of the several quotient figures by the divisor, together with the remainder, and the result, if the work is right, is like the dividend. See Example 2.

To prove Division by casting out the 9's, we find the excess of 9's in the divisor and also in the quotient, and multiply these excesses together and find the excess in their product. We then subtract the remainder from the dividend, and find the excess of 9's in the difference, which, if the work is right, will be equal to the excess found in the product of the excesses above named. See Example 3.

To prove Division by Division itself, we subtract the remainder from the dividend, and divide the difference by the quotient, and, if the work is right, the result will be equal to the original divisor. See Example 4.