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 prefix 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 say, 3 is in 12, 4 times.

16. Divide 56 by 4. 17. Divide 125 by 5.

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. Now 2 is in 16, 8 times.

21. Divide 2545 by 5.

22. Divide 3604 by 4.

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 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 annex 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 place 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 flure of the dividend, how proceed? When there is a remainder, after dividing the laat figure 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 ramainder 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, cau 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 division is mentioned?

24. A man having 600 dollars, wished to lay it out in flour, at 7 dollars a barrel: how many whole barrels could he buy, and how many dollars would he have left?

25. If you read 9 pages each day, how long will it take you to read a book through which has 828 pages ? 26. If I pay 8 dollars a yard for broadcloth, how many yards can I buy for 1265 dollars?

27. If a stage coach goes at the rate of 8 miles per hour, how long will it be in going 1560 miles?

28. If a ship sails 9 miles an hour, how long will it be in sailing to Liverpool, a distance of 3000 miles?

LONG DIVISION.

ART. 62. Ex. 1. A man having 156 dollars laid it out in sheep at 2 dollars apiece: how many did he buy?

Analysis.-Reasoning as before, since 2 dollars will buy 1 sheep, 156 dollars will buy as many sheep as 2 dollars are contained times in 156 dollars.

Directions. Having written the divisor on the left of the dividend as in short division, proceed in the following manner:

First. Find how many times the divisor (2) is contained in (15) the

Operation.

Divis. Divid. Quot 2) 156 (78

14.

16

16

first two figures of the dividend, and place the quotient figure (7) on the right of the dividend with a curve line between them. Second. Multiply the divisor by the quotient figure, (2 times 7 are 14,) and write the product (14) under the figures divided. Third. Subtract the

product from the figures divided. (The remainder is 1.) Fourth. Bringing down the next figure of the dividend, and placing it on the right of the remainder we have 16. Now 2 is contained in 16, 8 times; place the 8 on the right hand of the last quotient figure, and multiplying

the divisor by it, (8 times 2 are 16,) set the product under the figures divided, and subtract as before. Therefore 156 dollars will buy 78 sheep, at 2 dollars apiece.

63. When the result of each step in the operation is set down, the process of dividing is called LONG DIVISION. It is the same in principle as Short Division. The only difference between them is, that in Long Division the result of each step in the operation is written down, while in Short Division we carry on the whole process in the mind, simply writing down the quotient.

Note. To prevent mistakes, it is advisable to put a dot under each figure of the dividend, when it is brought down.

Solve the following examples by Long Division:

OBS. When the divisor is not contained in the first two figures of the dividend, find how many times it is contained in the first three, or the fewest figures which will contain it, and proceed as before.

9. How many times is 13 contained in 10519?

Thus, 13 is contained in 105, 8 times; set the 8 in the quotient then multiplying and subtracting, the remainder is 1. Bringing down the next figure we have 11 to be divided by 13. But 13 is not contained in 11;

Operation. 13)10519(809's Ans.

104

119

117

2 rem.

therefore we put a cipher in the quotient, and bring down the next figure. (Art. 57.) Then 13 is contained in 119,

QUEST.-63. What is long division? What is the difference between long and short division?