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Operation. We first inquire how many

Divisor 4 ) 9 48 Dividend, times 4, the divisor, is contained „ „ „ « . in 9, the first left-hand figure of

1 6' Quotient, the dividend, which is hundreds, and find it contained 2 times, and 1 hundred remaining. We write the 2 directly under 9, its dividend, for the hundreds' figure of the quotient. To 4, the next figure of the dividend, which is tens, we regard as prefixed the 1 hundred that was remaining, which equals 10 tens, and thus form 14 tens, in which we find the divisor 4 to be contained 3 times, and 2 tens remaining. We write the 3 for the tens' figure in the quotient, and the 2 tens that were remaining, equal 20 units, we regard as prefixed to 8, the last figure of the dividend, which is units, in which the divisor 4 is contained 7 times. Writing the 7 for the units' figure of the quotient, we have 237 as the entire quotient, equal the number of yards of cloth at 4 dollars a yard that can be bought for 948 dollars.

2. How many times does 3979 contain 17?

Ans. 234-j1,- times.

Operation. We say, 17 in 39,

Dividend. 2 times. The 2 we

Divisor 17)3979(234 T'T Quotient. write in the quo" '3 4 v TT ^ tient. 17X2 =

34, which we write

5 7 under the 39. 39

5 1 — 34 = 5, to which

bringing down the

6 9 next figure of the

6 8 dividend, we form

7 , j 57. 17 in 57, 3

1 Remainder. times Thewe

write in the quotient. 17X3 = 51, which we write under the 57. 57 — 51 = 6, to which bringing down the next figure of the dividend, we form 69. 17 in 69, 4 times. The 4 we write in the quotient. 17X4 = 68, which we write under the 69. 69 — 68 = 1, a remainder, or a part of the dividend left undivided. 1 divided by 17 = f, (Art. 68). The if we write in the quotient, and obtain as the answer required

234"- ...

In this illustration, to render the explanation the more concise, the naming of the denominations of the figures has been omitted.

When, as in the operation preceding the last, results only are written down, the method is called short division; and when, as in the last operation, the work is written out at length, it is called long division. The principle is the same in both cases. Hence the general

Rule. Beginning at the left, find hoiv many times the divisor is contained in the fewest figures of the dividend that will contain it, for the first quotient figure.

Multiply the divisor by this quotient figure, and subtract the product from the figures of the dividend used. With the remainder, if any, unite the next figure of the dividend.

Find how many times the divisor is contained in the number thus formed, and write the figure denoting the result at the right of the former quotient figure.

Thus proceed until all the figures of the dividend are divided

Note 1. — The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be as large as, or larger than, the divisor, it will show that there is an error in the work, and that th« quotient figure should be increased.

Note 2. — If at any time the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished.

Note 3. — It will often happen that, when a figure of the dividend is taken, the number will not contain the divisor; and, in that case, a cipher must be placed in the quotient, and another figure of the dividend taken, and so on, until the number is large enough to contain the divisor.

Note 4. — If there be a remainder after dividing the last figure of the dividend, write it with the divisor underneath, with a line between them, at the right of the quotient.

74 i First Method of Proof. — Multiply the divisor by the quotient, and to the product add the remainder, if any, and if the work be right, the sum thus obtained will be equal to the dividend.

Note. — This method follows from division being the reverse of multiplication. (Art. 72.)

75i Second Method of Proof. — Find the excess of nines in the divisor, quotient, and remainder. Multiply the excess of nines in the divisor and quotient together, and to the product add the excess of nines in the remainder. If the excess of nines in this sum equal the excess of nines in the dividend, the work may be supposed to be right.

76. Third Method of Proof. — Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work be right.

77i Fourth Method of Proof. — Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work be right.

Note. — The first method of proof (Art. 74) is usually most convenient, and is most commonly employed.

Examples.

3. Divide 9184 by 7.

OPERATION,

Divisor 7)9184 Dividend.

1312 Quotient.

4. Divide 18988 by 759.

Ans. 1312.

PROOF BY MULTIPLICATION.

1312 Quotient.

7 Divisor. 9184 Dividend.

Ans. 25fyV.

PROOF BY THE NINES.

Divisor = 3 excess.

OPERATION.

Dividend.

Divisor 759)18988(25 Quotient. Quotient == 7 excess.

1518 Remainder = 4 excess.

3808

37 95 (3 X 7) + 4 = 7 excess.

1 3 Remainder Dividend = 7 excess.

5. Divide 147856 by 97. Ans. 1524|f

OPERATION.

Dividend.

Divisor 97)147856(1524 Quotient. +9 7

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OPERATION.

Dividend.

PROOF BY DIVISION.

Dividend.

Divisor 28 5)84645(297 Quotient 297)84645(285 Divisor

5 7 0 5 9 4

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7. 8. 9.

3)67856336 7)178985 11)1667789

22618778$

10. 11. 12.

5)334652 8)96723578 9)186731

13. 14. 15.

17)678916( 35)9106013( 91)6210011(

Quotients. Rem.

16. Divide 671678953 by 6. 111946492 1.

17. Divide 166336711 by 7. 23762387 2.

18. Divide 161331793 by 8. 20166474 . 1.

19. Divide 161677678 by 9. 17964186 4.

20. Divide 363895678 by 11. 33081425 3.

21. Divide 164378956 by 12. 13698246 4.

22. Divide 78950077 by 3. 1.

23. Divide 678956671 by 4. 3.

24. Divide 667788976 by 5. 1.

25. Divide 777777777 by 6. 3.

26. Divide 888888888 by 7. 6.

27. Divide 789636 by 46.

28. Divide 7967848 by 52. 153227 44.

29. Divide 16785675 by 61. 275175

30. Divide 675753 by 39.

31. Divide 5678911 by 82. 1.

32. Divide 6716394 by 94. 71451

33. Divide 1167861 by 135. 8650 111.

34. Divide 7861783 by 87. 90365 28.

35. Divide 1678567 by 365. 4598 297.

36. Divide 87635163 by 387. 226447 174.

37. Divide 34567890 by 6789. 5091 5091.

38. What is the value of 213255467083 H- 30204? 4207.

39. What is the value of 395020613 -r- 4444? 2341.

40. What is the value of 7207276639 -~ 9009? 4567.

41. What is the value of 454115186870257 -T- 500123 ? 8765.

42. How many barrels of flour, at 9 dollars a barrel, can be bought for 18621 dollars?

43. How much sugar at 15 dollars a hundred may be bought for 405 dollars?

44. A tailor has 938 yards of broadcloth; how many cloaks can be made of the cloth, if it require 7 yards to make one cloak?

45. What number multiplied by 1728 will produce 1705536?

Ans. 987.

46. A. Hartmann has sold his wagon to J. Herr for 85 dol. lars. He is to receive his pay in wood at 5 dollars a cord. How many cords will it require to pay for the wagon?

Ans. 17 cords.

47. The Bible contains 31,173 verses; how many must be read each day, that the book may be read through in a year of 365 days? Ans. 85^J| verses.

48. A train on the Liverpool Railroad runs at the rate of 65 miles an hour; how long would it take at that velocity to pass round the earth, the distance being about 25,000 miles?

Ans. 384T\ hours.

49. A gentleman possessing an estate of 66,144 dollars, bequeathed one fourth to his wife, and the remainder was divided among his 4 children; what was the share of each?

Ans. 12,402 dollars.

50. If the dividend is 6756785 and the quotient 193051, what is the divisor? Ans. 35.

51. A's age multiplied by 17, or B's age multiplied by 19, is equal to 1292 years, and the sum of their" ages is equal to G's age multiplied by 3. What is the age of each?

Ans. A's 76 years; B's 68 years; C's 48 years.

78. When the divisor is a composite number.

Ex. 1. A farmer bought 21 horses for 2625 dollars; how many dollars did each cost? Ans. 125 dollars.

Operation. The factors of 21 are

3) 2 6 2 5 dolls., cost of 21 horses. 3 and 7. Now, if we

7 )875 dolls., cost of 7 horses. «»£*5

12 5 dolls., cost of 1 horse. 3, wo obtain 875 dollars,

the coot of 7 horses, since there are 7 times 3 in 21. Then, dividing the 875 dollars, the cost >»f 7 horses, by 7, we obtain 125 dollars, the cost of 1 horse.

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