OBS. The object in each of the last twenty questions, is to divide a given number into several equal parts, and ascertain the value of these parts; but the method of solving them is precisely the same as that of the preceding ones.

64. The process by which the foregoing examples are solved, is called DIVISION.

It consists in finding how many times one given number contains another.

The number to be divided is called the dividend.

The number by which we divide is called the divisor. The number obtained by division, or the answer to the question, is called the quotient. It shows how many times the dividend contains the divisor.

OBS. The term quotient is derived from the Latin word quoties, which signifies how often, or how many times. Hence, it may be said,

65. Division is finding a quotient, which multiplied into the divisor, will produce the dividend.

66. The number which is sometimes left after division, is called the remainder. Thus in the twenty-first example, when we say 4 is contained in 25, 6 times and 1 over, 4 is the divisor, 25 the dividend, 6 the quotient, and 1 the remainder.

OBS. 1. The remainder is always less than the divisor; for if it were equal to, or greater than the divisor, the divisor could be contained once more in the dividend.

2. The remainder is also of the same denomination as the dividend; for it is a part of it.

67. Division is denoted in two ways:

QUEST.-64. What is the operation called by which the above examples have been solved? In what does it consist? How many numbers are given? What is the number to be divided called? The number by which we divide? What is the number obtained, called? What does the quotient show? 65. What then may division be said to be? 66. What is the number called which is sometimes left after division? When we say 4 is in 25, 6 times and 1 over, what is the 4 called? The 25? The 6? The 1? When we say 6 is in 45, 7 times and 3 over, which is the divisor? The dividend? The quotient? The remainder. Obs. Is the remainder greater or less than the divisor? Why? Of what denomination is it? Why? 67. How many ways is division denoted?

First, by a horizontal line between two dots, called the sign of division, which shows that the number preceding it is to be divided by the number after it. Thus the expression 24÷6, signifies that 24 is to be divided by 6.

Second, division is often expressed by placing the divisor under the dividend with a short line between them. Thus the expression 35, shows that 35 is to be divided by 7, and is equivalent to 35÷7.

OBS. 1. It will be perceived that division is similar in principle to subtraction, and may be performed by it. For instance, to find how many times 3 is contained in 12, as in the first example, subtract 3 (the divisor) continually from 12 (the dividend) until the latter is exhausted; then counting these repeated subtractions, we shall have the true quotient. Thus 3 from 12 leaves 9; 3 from 9 leaves 6; 3 from 6 leaves 3; 3 from 3 leaves 0. Now by counting, we find that 3 can be taken from 12, 4 times; or that 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. 2. 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.

3. When the dividend denotes things of one denomination only, the operation is called Simple Division.

EXERCISES FOR THE SLATE.

Ex. 1. How many barrels of cider, at 2 dollars a barrel, can you buy for 648 dollars?

Suggestion. Since 2 dollars will buy 1 barrel, 648 dollars will buy as many barrels as 2 is contained times in 648.

QUEST. What is the first? What does this sign show? What is the second way of denoting division? Obs. To what rule is division simiJar in principle? How is division sometimes defined? Of what is di vision the reverse? How does this appear? When the dividend denotes things of one denomination only, what is the operation called?

Operation.

Divisor. Dividend.

2) 6 4 8

Quot. 3 2 4

Having written the numbers upon the slate, as in the margin, we proceed thus: 2 is contained in 6, 3 times. Now as the 6 denotes hundreds, the 3 must also be hundreds. We therefore write it in hundreds' place; that is, under the figure which we are dividing. 2 in 4, 2 times. Since the 4 is tens, the 2 must also be tens, and we write it in tens' place. 2 in 8, 4 times. The 8 is units; hence the 4 must be units, and we write it in units' place. The answer is 324 barrels.

7. How many hats, at 2 dollars apiece, can be bought for 468 dollars? Ans. 234 hats.

8. How many sheep, at 3 dollars a head, can be bought for 369 dollars?

9. A man wished to divide 248 acres of land equally between his two sons: how many acres will each receive? 10. How many times is 4 contained in 488?

68. Hence, when the divisor contains but one figure, Write the divisor on the left hand of the dividend with a curve line between them; then, beginning at the left hand, divide each figure of the dividend by the divisor, and set each quotient figure directly under the figure from which it

arose.

11. A farmer bought 96 dollars worth of dry goods, and agreed to pay in wood at 3 dollars a cord: how many cords will it take to pay his bill? Ans. 32 cords.

12. In 963 feet, how many yards are there, allowing 3 feet to a yard?

16.

13. Divide 63936 by 3. 14. Divide 48848 by 4. 15. Divide 55555 by 5. Divide 2486286 by 2. 69. When the divisor is not contained in the first

QUEST.-68. How do you write the numbers for division? Where begin to divide Where place each quotient figure? 69. When the divisor is not contained in the first figure of the dividend, what must be done?

figure of the dividend, we must find how many times it is contained in the first two figures.

17. How many hats, at 3 dollars apiece, can be bought for 249 dollars?

3)249

Operation. Since the divisor (3) is not contained in (2) the first figure of the dividend, we say, 3 is in 24, 8 times, and write the 8 under the 4. 3 in 9, 3 times.

Quot. 83

18. Divide 124 by 4. 20. Divide 255 by 5. 22. Divide 24693 by 3. 24. Divide 35555 by 5. 26. Divide 64888 by 8.

Ans. 83 hats. 19. Divide 366 by 6. 21. Divide 1248 by 4. 23. Divide 4266 by 6. 25. Divide 5677 by 7. 27. Divide 8199 by 9.

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.

OBS. 1. To prefix means to place before or at the left hand.

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

Operation. 3)741

When we divide 7 by 3, there is 1 remainder. This we prefix mentally to the next figure of the dividend. Ans. 247 acres. We then say, 3 in 14, 4 times, and 2 over. Prefixing the remainder 2 to the next figure, as before, we say, 3 in 21, 7 times.

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?

QUEST. 70. If there is a remainder after dividing a figure of the dividend, what must be done with it? Obs. What does the word prefix mean? When the divisor is not contained in a figure of the dividend, what must be done?

Operation.
6)642

OBS. 2. In this example the divisor is not contained once in the tens' figure of the dividend; we must therefore write a cipher in the quotient, and prefix the 4 to the next figure of the dividend, as if it were a remainder. We then say, 6 in 42, 7 times, and place the 7 under the 2.

36. Divide 36060 by 6.

38. Divide 45900 by 9.

Ans. 107 barrels.

37. Divide 49000 by 7. 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.

43. Divide 30420 by 6.

44. Divide 25105 by 5.

45. Divide 643240 by 8.

46. A merchant wished to divide 49 oranges equally among 4 boys: how many must he give to each ?

Operation. After giving them 12 apiece, it will be seen that there is one re4)49 mainder, or 1 orange left, which Ans. 12-1 remainder. 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, (Art. 67,) and in order to complete the quotient, the must be annexed to the 12. The true quotient, therefore, is 12 and 1 divided by 4, and should be written thus, 124. Hence,

71. 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.

47. A shoemaker has 375 pair of boots, which he wishes to pack in 6 boxes: how many pair can be put in a box? Ans. 623.

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

QUEST-71. When there is a remainder, after dividing the last figure of the dividend, what must be done with it?