been taken from 12, 4 times; consequently 3 is contained in 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 denomina tion only, the operation is called Simple Division.

QUEST. Obs. When the dividend denotes things of one denomination only

what is the operation called?

SHORT DIVISION.

ART. 54. Ex. 1. How many yards of cloth, at 2 dollars 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 n 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 igure 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?

Operation.

2)124

Ans. 62 bu.

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

The an

11. Divide 142 by 2.

12. Divide 129 by 3.

13. Divide 1648 by 4.

14. Divide 2877 by 7.

56. After dividing any figure of the dividend, if there is a remainder, prefix it mentally to the next figure of the dividend, and then divide this number as before.

Note.-To prefir means to place before, or at the left hand.

15. A man bought 42 peaches, which he divided equally among his 3 children: how many did he give to each?

When we divide 4 by 3, there is 1 remainder. This we prefix mentally to the next figure of the dividend. We then 3 is in 12, 4 times.

16. Divide 56 by 4.

17. Divide 125 by 5.

say,

Operation. 3)42

14 Ans.

18. Divide 456 by 6. 19. Divide 3648 by 8.

57. Having obtained the first quotient figure, if the divisor is not contained in any figure of the dividend, place a cipher in the quotient, and prefix this figure to the next one of the dividend, as if it were a remainder.

20. If hats are 2 dollars apiece, how many can be bought for 216 dollars?

As the divisor is not contained in 1, the second figure of the dividend, we put a 0 in the quotient, and prefix the 1 to the 6 as directed above.

is in 16, times.

21. Divide 2545 by 5.

22. Divide 3604 by 4.

Now 2

Operation.

2)216

Ans. 108 hats.

23. Divide 6402 by 6.

24. Divide 4024 by 8.

25. A man divided 17 loaves of bread equally between 2 poor persons: how many did he give to each?

Suggestion.-Reasoning as before, he gave each as many loaves as 2 is contained times in 17.

Operation.
2)17

Quot. 8-1 remainder.
Ans. 8 loaves.

Thus, 2 is contained in 17, 8 times and 1 over; that is, after giving them 8 loaves apiece, there is one loaf left which is not divided. Now 2 is not contained in 1; hence the division must be represented by writing the 2 under the 1, thus, (Art. 52,) which must be annexed to the 8. The true quotient, is 8. He therefore gave eight and a half loaves to each. Hence,

58. When there is a remainder after dividing the last figure of the dividend, it should always be written over the divisor and annexed to the quotient.

Note.-To anner means to place after, or at the right hand.

59. When the process of dividing is carried on in the mind, and the quotient only is set down, the operation is called SHORT DIVISION.

60. From the preceding illustrations and principles, we derive the following

RULE FOR SHORT DIVISION.

I. Write the divisor on the left of the dividend, with a curve line between them.

Beginning at the left hand, divide each figure of the dividend by the divisor, and place each quotient figure under the figure divided.

II. When there is a remainder after dividing any figure, prefix it to the next figure of the dividend and divide this number as before. If the divisor is not contained in

QUEST.-59. What is Short Division? 60. How do you write numbers for short division? Where begin to divide ? Where placo each quotient figure? When there is a remainder after dividing a figure of the dividend, what must be done with it? If the divisor is not contained in a figure of the dividend, how proceed? When there is a remainder, after dividing the last Aguro of the dividend, what must be done with it?

any figure of the dividend, place a cipher in the quotient, and prefix this figure to the next one of the dividend, as if it were a remainder. (Arts. 56, 57.)

III. When there is a remainder after dividing the last figure, write it over the divisor and annex it to the quotient.

61. PROOF.-Multiply the divisor by the quotient, to the product add the remainder, and if the sum is equal to the dividend, the work is right.

OBS. Division may also be proved by subtracting the remainder, if any, from the dividend, then dividing the result by the quotient.

19. How many pair of shoes, at 2 dollars a pair, can you buy for 126 dollars?

20. How many hats, at 4 dollars apiece, can be bought for 168 dollars?

21. A man bought 144 marbles which he divided equally among his 6 children: how many did each receive?

22. A man distributed 360 cents to a company of poor children, giving 8 cents to each: how many children were there in the company ?

23. How many yards of silk, at 6 shillings per yard, can I buy for 450 shillings?

QUEST.-61. How is division proved? Obs. What other way of proving divizion is mentioned?