17. Give the quotients of the numbers in the following line divided by 2; by 4; by 6; by 8; by 10.

8, 12, 16, 20, 24, 32, 36, 48, 60, 64.

18. Give the quotients of the numbers in the following line divided by 3; by 5; by 7; by 9; by 8.

28, 33, 35, 42, 49, 63, 81, 96, 100.

Supply the necessary numbers in the following:

23. In 24 how many times 2? 6? 8? 4? 12?

24. In 45 how many times 5? 3? 9? 15?

25. In 60 how many times 5? 10? 6? 4? 12? 15? 20? 26. At 8 cents each, how many cocoa-nuts can be bought for 88 cents?

27. If a dozen of eggs cost 48 cents, what does one cost? 28. How many pounds of meat can be bought for 72 cents at 8 cents a pound?

29. How many yards of cloth at $6 a yard will pay for 8 tons of coal at $9 a ton?

30. If 4 men do a piece of work in 12 days, how long will it take 6 men?

31. What number divided by 9 will give 6?

32. If 4 cords of wood cost $ 36, what will 7 cords cost?

33. How many twelve-quart cans will hold 84 gallons of milk?

34. If a train of cars runs 108 miles in 9 hours, what is the rate per hour?

71. Principles of Division.

1. The dividend is the product of the divisor and the quotient.

2. When the divisor and the dividend are like numbers, the

quotient will be an abstract number.

3. When the divisor is an abstract number, the dividend and the quotient will be like numbers.

WRITTEN EXERCISES.

35. Divide 8574 by 6.
Divisor 6) 8574 Dividend.

1429 Quotient.
6

Proof

8574

Solution. - We write the divisor at the left of the dividend, with a curve between them, and begin at the left to divide.

6 in 8 thousands, 1 thou

sand times, with 2 thousands, or 20 hundreds, remaining. We write the 1 thousand beneath in thousands' place.

Uniting the 20 hundreds with the 5 hundreds of the dividend, we have 25 hundreds. 6 in 25 hundreds, 4 hundred times, with 1 hundred, or 10 tens, remaining. We write the 4 hundreds beneath in hundreds' place.

Uniting the 10 tens with the 7 tens of the dividend, we have 17 tens. 6 in 17 tens, 2 tens times, with 5 tens, or 50 ones, remaining. We write the 2 tens beneath in tens' place.

Uniting the 50 ones with the 4 ones of the dividend, we have 54 ones. 6 in 54 ones, 9 times. We write the 9 ones beneath in ones' place.

The quotient is 1 thousand 4 hundreds 2 tens 9 ones, or 1429.

To prove the work, we multiply the quotient by the divisor, and have as the product the dividend (Art. 71).

53. How many tons of coal at $5 a ton can be bought for $2025?.

54. How far must a vessel sail each day to sail 1438 miles in 7 days?

55. Paid for 8 yards of broadcloth $43.60. What was the cloth a yard?

56. If the dividend is 20818, and the divisor 9, what is the quotient?

ORAL EXERCISES.

57. How many times is of 32 contained in of 60?

58. The product of two numbers is 56, and one of the numbers 8. What is the other number?

59. If the dividend is 72, and the divisor 9, what is the quotient?

60. Mr. Jones can earn in a month $81, and his son one ninth as much. How much can his son earn?

61. A farm of 120 acres has been divided into 12 lots. How many acres in each lot?

of 110 ?

of 96?

In what

62. What is of 64? of 81 ? 63. One man can do a piece of work in 132 days. time can 11 men do it?

64. A man earned $144 in 12 weeks. What was that a week?

65. How many 10's in 100? How many 100's in 1000? 66. In 65 how many tens and what over? In 235 how many hundreds and what over?

67. How many times 20 in 100? How many times (204) in (1004)? How many times (20 × 2) in (100 × 2)?

72. A General Principle of Division.

Dividing or multiplying both dividend and divisor by the same number does not alter the quotient.

12

with these 10 tens the 5 ones of the dividend, we have 105 ones. is in 105, 8 times. We write the 8 ones above the ones of the dividend.

Writing

8 times 12 are 96, which taken from the 105 of the dividend leaves 9, and indicating its division by the divisor, we have theas a part of the quotient, we have as the quotient 98.

69. Divide 9876 by 21. 70. Divide 3276 by 14. 71. Divide 31278 by 23.

72. Divide 106.97 by 19.

73.

Divide 26.575 by 25. 74. Divide 1868.5 by 37.

73. Rule for Division.

Write the divisor at the left of the dividend with a curve between them.

Find how many times the divisor is contained in the few

est left-hand figures of the dividend that will contain it, and write the result under or over the right-hand figure of the dividend used, as the first quotient figure.

Multiply the divisor by this quotient, subtract the product from the partial dividend used, and to the remainder annex the next dividend figure for a second partial dividend.

Divide and proceed as before, until all the dividend figures have been used.

If there is a final remainder, write it with the divisor under it, as a part of the quotient.

NOTE 1. - The division is called Short Division when only the divisor, dividend, and quotient are written; and Long Division when each process of the solution is written.

In short division the quotient is usually written below the dividend, and in long division over, or at the right of, the dividend.

NOTE 2. When the divisor is an integer, and the quotient is written under 'or over the dividend, the decimal point in the quotient must be placed immediately under or over the point in the dividend.

74. PROOF. Multiply the quotient by the divisor, and to the product add the remainder, if any. The result should be the dividend.

75. Divide 182.72 by 45, and prove the work.

76. Find the quotient of $114.48 divided by 36.

77. Find one twenty-eighth of $ 896.56.