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70. 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.

If the divisor is not contained in any figure of the dividend, place a cipher in the quotient, and prefixing this figure to the next one in the dividend, proceed as before.

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

OBS. 1. The learner will observe, in division we begin at the left hand, instead of the right, as in Addition, Subtraction, and Multiplication. The reason is, that in dividing a higher order, there is frequently a remainder which must be united with the next lower order, before the division can be performed.

2. The reason for placing a cipher in the quotient, when the divisor is not contained in a figure of the dividend, is to preserve the true local value of the several quotient figures.

3. The divisor is placed on the left of the dividend, and the quotient under it, merely for the sake of convenience. When division is represented by the sign (+), the divisor is placed on the right of the dividend; and when represented in the form of a fraction, the divisor is placed under the dividend.

17. How many hats, at 3 dollars apiece, can be bought for 8421 dollars?

Operation.

3)8421 Ans. 2807 hats.

Suggestion.-Dividing 8 by 3, there is 2 remainder. This we prefix mentally to the next figure of the dividend. Now, 3 is in 24, 8 times. Again, 3 is not contained in 2, the next figure of the dividend; we therefore place a cipher in the quotient, and prefixing the 2 to the 1, divide as before.

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

18. A teacher having 125 apples, wishes to divide them equally among 4 pupils: how many can he give to each?

Operation.
4)125

31-1 rem.

Ans. 31 apples.

Suggestion. After giving them 31 apiece, it will be seen that there is one remainder, or 1 apple left, which is not divided. Now it is plain that the whole dividend must be divided, in order to render the division complete. But 4 is not contained in 1; hence the division must be represented by writing the 4 under the 1, thus, and in order to complete the quotient, the must be annexed to the 12. (Art. 67.) The true quotient, therefore is 121, which is read, "twelve and one fourth."

72. 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. (Art. 68.)

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 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.

III. When there is a remainder after dividing the last figure, write it over the divisor and annex it to the quotient. (Art. 71.)

PROOF.-Multiply the divisor by the quotient, to the product add the remainder, and if the result is equal to the dividend, the work is right. (Art. 65.)

19. Divide 387475 by 6, and prove the operation.

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OBS. 1. The reason of this method of proof, may be seen from the fact that the quotient shows how many times the divisor is contained in the dividend; consequently, if the divisor is repeated or taken as many times as there are units in the quotient, it must produce the dividend. (Art. 64.)

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

Note.-As soon as the learner becomes familiar with the process of dividing by Short Division, he should drop the intervening words as in the preceding

QUEST.-72. How do you write numbers for division? How proceed in Short Division? When there is a remainder after dividing a figure, what do you do with it If the divisor is not contained in any figure of the dividend, how proceed? Note. What is the meaning of the term, prefix? When there is a remainder after dividing the last figure of the dividend, what must be done with it? 70. Obs. Why place the divisor on the left of the dividend and the quotient under it? Why begin to divide at the left hand? Why place a cipher in the quotient, when the divisor is not contained in a figure of the dividend? How is division proved? Obs. What other way of proving division is mentioned ?

rules, simply pronouncing the quotient figures and setting them down at once. Thus, in the example above instead of saying, 6 is contained in 38, 6 times and 2 over; 6 is contained in 27, 4 times and 3 over, &c., he should learn to say, siz, four, five, seven, &c., setting down each quotient figure while pronouncing it.

20. Divide 255 by 5. 22. Divide 24693 by 3. 24. Divide 35555 by 5.

26. Divide 64888 by 8.

21. Divide 1248 by 4.

23. Divide 4266 by 6.

25. Divide 5677 by 7.

27. Divide 8199 by 9.

28. A man bought 741 acres of land, which he divided equally among his 3 sons: how many acres did each receive?

29. If a man travel at the rate of 5 miles an hour, how long will it take him to travel 345 miles? Ans. 69 hours.

30. If 192 pounds of flour were equally divided among 4 persons, how many pounds would each receive?

31. Divide 45690 by 6.

33. Divide 81670 by 5.

32. Divide 52584 by 8.

34. Divide 28296 by 9.

35. When flour is 6 dollars a barrel, how much can be bought

for 642 dollars?

36. Divide 36060 by 6.

37. Divide 49000 by 7.

38. Divide 45900 by 9.

39. Divide 568000 by 8.

40. Allowing 5 yards of cloth for a suit of clothes, how many suits can be made from 1525 yards? Ans. 305 suits.

41. A company of 3 men agree to pay a bill of 321 dollars: how many dollars must each man pay?

42. Divide 14350 by 7. 44. Divide 25105 by 5.

43. Divide 30420 by 6.

45. Divide 643240 by 8.

46. A merchant wishes to divide 549 oranges equally among 4 boys: how many must he give to each?

47. A shoemaker has 372 pair of boots, which he wishes to pack in 6 boxes: how many pair can he put into a box?

48. A baker wishes to lay out 756 dollars in flour: how much can he buy, when the price is 5 dollars a barrel?

49. How many yearlings, at 9 dollars a head, can be bought for 468 dollars?

50. How many acres of land, at 6 dollars an acre, can I buy for 973 dollars?

51. Divide 5468053 by 7. 53. Divide 6000000 by 9. 55. Divide 7034016 by 10. 57. Divide 8306734 by 12.

52. Divide 4672304 by 8. 54. Divide 7003041 by 6. 56. Divide 8097603 by 11. 58. Divide 9603405 by 12.

LONG DIVISION.

ART. 73. Long Division is the process of dividing, when the result of each step in the operation is set down.

74. Long Division is the same in principle as Short Division. The only difference between them is, that in the former, the result of each step in the operation is set down; while in the latter, the process of dividing is carried on in the mind, and the quotient only is set down. (Art. 68.)

Operation.

Divi. Divid. Quot.

4)1504(376 12

30

28

24

24

Ex. 1. Divide 1504 by 4, using Long Division. Suggestion. Having set down the numbers as in Short Division, we first find how many times the divisor 4, is contained in 15, the fewest figures on the left of the dividend that will contain it, (4 is in 15, 3 times,) and place the quotient figure on the right of the lividend with a curve line between them. Next, we multiply the divisor by the quotient figure, (3 times 4 are 12,) and write the product under the figures divided. We then subtract this product from the figures divided. (12 from 15 leaves 3.) Finally, we bring down the next figure of the dividend, and placing it on the right of the remainder, divide this number as above. (4 is in 30, 7 times.) Place the 7 on the right of the last quotient figure, then multiply, subtract, and proceed to find the next figure of the quotient as before.

75. From the preceding operation, the learner will per ceive, there are four steps in Long Division: 1st. Find how many times the divisor is contained, &c.; 2d. Multiply; 3d. Subtract; 4th. Bring down.

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

2. Divide 578 by 2, and prove the operation. Ans. 289.

3. Divide 7560 by 5. Ans. 1512.

4. Divide 126332 by 4. Ans. 31583.

5. How many times is 6 contained in 763251 ?

QUEST.-73. What is Long Division? 74. What is the difference between Long and Short Division? 75. How many steps are there in Long Division?

6. How many times is 3 contained in 4026942? 7. How many times is 8 contained in 2612488? 8. How many times is 5 contained in 1682840? 9. How many times is 7 contained in 45063284? 10. How many times is 9 contained in 650031507? 11. Divide 2234 by 21.

Suggestion. Having subtracted the 21, and brought down the next figure of the dividend, we have 13 to be divided by 21. But 21 is not contained in 13; we therefore put a cipher in the quotient, and bring down the next figure. 21 is in 134, 6 times, and 8 rem.

Then

Write

Operation. 21)2284(106

21

134

126

8 rem.

the 8 over the divisor, and annex it to the quotient. Hence,

76. After the first quotient figure is obtained, for each figure of the dividend which is brought down, either a significant figure, or a cipher must be put in the quotient.

77. From the preceding illustrations and principles we derive the following

RULE FOR LONG DIVISION.

I. Beginning on the left of the dividend, find how many times the divisor is contained in the fewest figures that will contain it, and place the quotient figure on the right of the dividend with a curve line between them.

II. Multiply the divisor by this figure and subtract the product from the figures divided; to the right of the remainder bring down the next figure of the dividend, and divide this number as before. Proceed in this manner till all the figures of the divi» dend are divided.

III. When there is a remainder after dividing the last figure, vorite it over the divisor, and annex it to the quotient, as in short division. (Art. 71.)

OBS. 1. To find the first quotient figure when the divisor is large, the learner should consider how many times the first figure of the divisor is contained in the

QUEST.-77. What is the rule for Long Division? If there is a remainder after dividing the last figure of the dividend, what must be done with it? Obs When the divisor is large, how do you find the first quotient figure?

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