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larger sums into smaller ones; the smaller being equal one to another; as twelve, divided into four equal parts, gives us three for each part; and also shows us that three can be substracted from twelve four times.

EXAMPLES

For Exercise on a Slate.

1. If you can buy one bushel of wheat for 2 dollars; how many bushels can you buy for 628 dollars? Ans. 314 bushels.

EXPLANATIONS.

In this example, 628, Divisor 2] 628 dividend. the larger number, is the

dividend, and the 2 is

314 quotient. You must begin You must set

placed at the left for a divisor. with the 6, and say, 2 in 6, 3 times. down the 3 directly under the 6; and proceeding to the next figure, you must say, 2 in 2, 1 time. You must set down the 1 under the 2; and proceeding to the next figure, you must say, 2 in 8, 4 times. You must set down the 4 under the 8, and then the work is done. Thus you will perceive, that 2 is contained 314 times in 628; and, therefore, 314 is the quotient,

or answer.

PROOF.

2 divisor.

You may prove Division by Mul- 314 quotient. tiplication. Multiply the quotient by the divisor. If there be a remainder, you must add it in with the

628 dividend.

product of the quotient, multiplied by the divisor If the product of the quotient, multiplied by the divisor, be like the dividend, the work is right. The principle of proving Division by Multiplication is very plain; for, as Division is the separating of larger numbers into smaller ones, so is Multiplication the bringing together of smaller numbers into one sum. Thus, 628, divided by 2, is separated into two smaller parts, each part 314; and, consequently, if you multiply it by 2, you bring the two parts together, by increasing it two times.

2. A man divided 472 cents between 4 boys; how many cents did each boy receive? Ans. 118 cents.

EXPLANATIONS.

In this example, you Divisor 4] 472 dividend. must begin with the 4, and say, 4 in 4, 1 time.

118 quotient. You must set down the 1 directly under the 4; and proceeding to the next figure, you must say, 4 in 7, 1 time and three over. You must set down the 1 under the 7, and carry the 3 as so many tens; and then say, 4 in 32, 8 times. You must set down the 8 under the 2, and then the work is done.

The reason for calling every one that is over, ten, when you divide, is very evident; for one in the left hand row, or column, of figures is always equal to ten in the next place to the right; and, therefore, you must always call the remainder tens as long as there are figures at the right hand in the dividend.

3. A merchant bought a quantity of flour at 5 dollars a barrel, and paid for the whole 8605 dollars; how many barrels did he buy? Ans. 1721 barrels.

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EXPLANATIONS.

Beginning with the 8, Divisor 5) 8605 dividend. in the place of thousands, you must say, 5 in

1721 quotient. 8, 1 time, that is, 5 is in 8000, 1000 times and three over, that is, 3000 over; and you must carry this to the next figure, the place of hundreds, that is, thirty hundreds, and add it to the 6 hundreds, which will make the number 36, that is, 36 hundreds; and you must then say, 5 in 36, 7 times, that is, 5 is in 3600, 700 times and one over, that is, 100 over; and you must carry this to the next figure, the place of tens, that is, one hundred, and add it to the cipher, in the place of tens, which will make the number no larger, that is, only one hundred, or ten tens; and you must then say, 5 in 10, 2 times, that is, 5 in 100, 20 times, and none over; and you must then say, 5 in 5, 1 time. You must set down the 1 under the 5, and then the work is done. By paying particular attention to the preceding EXPLANATIONS, you will be able to work any sum in Division; that is, any sum in which the divisor does not exceed 12; for you must continually bear in mind, that the object of these EXPLANATIONS is to explain to you the principles upon which the rules are founded.

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(5.)

Divisor 2)394968 div. Divisor 3)572859 div.

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15. A merchant paid 1125 cents for a quantity of lemons, at 5 cents apiece; how many lemons did he buy? Ans. 225 lemons.

16. A farmer sold a quantity of pork for 3000 cents, at 8 cents a pound; how many pounds did he sell? Ans. 375 pounds.

17. A farmer sold a quantity of cheese for 2982 cents, at 7 cents a pound; how many pounds did he sell? Ans. 426 pounds.

18. A man bought a farm for 1878 dollars, at 3 dollars an acre; how many acres did he buy? Ans. 626 acres.

19. How many oranges can you buy for 4448 cents, if you pay 8 cents apiece? Ans. 556 oranges. 20. A drover bought 12 oxen for 636 dollars; how many dollars did he pay for each? Ans. 53 dollars.

21. If a man spend 984 dollars in 12 months; how many dollars does he spend in each month? Ans. 82 dollars.

22. There were 5 men who traded and gained 4900 dollars; how many dollars were each man's share? Ans. 980 dollars.

23. There were 4 merchants who sold a quantity of goods for 11516 dollars; how many dollars were each man's share? Ans. 2879 dollars.

EXPLANATIONS.

Thus, I have treated of the manner in which Division is carried on in the more simple operations in which it is employed; that is, in the division of any sum by any number that does not exceed 12. In every process of this sort, the operation is performed in a single line, as it is in Multiplication, when the multiplier does not exceed 12. When the

divisor does not exceed 12, the operation is called SHORT DIVISION. When the divisor exceeds 12, the operation is called LONG DIVISION. On this I shall now proceed to treat.

Q. When the divisor exceeds 12, where must it be placed?

A. It must be placed at the left of the dividend as before.

Q. Where must the quotient be placed?

A. It must be placed at the right of the dividend.

Q. When the divisor exceeds 12, how many figures of the dividend must first be taken?

A. As many figures at the left hand must be taken as will contain the divisor once or more.

Q. After having ascertained how many times the divisor is contained in the figures taken, how do you proceed?

A. The divisor must be multiplied by the quotient figure, and the product placed under the figures of the dividend which were taken. This product must then be substracted from the

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