104. 5. A certain cornfield contains 2688 hills of corn planted in rows, which are 56 hills long, how many rows are there?

56) 2688 ( 48 224

448 448

Here, as 56 is not contained in 26, it is necessary to take three figures, or 268, for the first partial dividcnd; but there may be some difficulty in finding how many times the divisor may be had in it. It will, however, soon be seen by inspection, that it cannot be less than 4 times, and by making trial of 4, we find that we cannot have a larger number than that in the ten's place of the quotient, because the remainder, 44, is less than 56, the divisor. In multiplying the divisor by the quotient figure, if the product be greater than the part of the dividend used, the quotient figure is too great; and in subtracting this product, if the remainder exceed the divisor, the quotient figure is too small; and in each case the operation must be repeated until the right figure be found.

SIMPLE DIVISION.

DEFINITIONS.

105. Simple Division is the method of finding how many times one simple number is contained in another; or, of separating a simple number into a proposed number of equal parts. The number which is to be divided, is called the dividend; the number by which the dividend is to be divided, is called the divisor; and the number of times the divisor is contained in the dividend, is called the quotient. If there be any thing left after performing the operation, that excess is called the remainder, and is always less than the divisor, and of the same kind as the dividend.

RULE.

106. Write the divisor at the left hand of the dividend; find how many times it is contained in as many of the left hand figures of the dividend, as will contain it once, and not more than nine times, and write the result for the highest figure of the quotient. Multiply the divisor by the quotient figure, and set the product under the part of the dividend used, and subtract it therefrom. Bring down the next figure of the dividend to the right of the remainder, and divide this number as before; and so on till the whole is finished.

NOTE. If after bringing down a figure to the remainder, it be still less than the divisor, place a cipher in the quotient, and bring down another figure. (103.) Should it still be too small, write another cipher in the

What is Simple Division?

What is meant by the dividend? by the divisor? by the quotient? by the remainder?

Of what kind is the remainder?

How may the division of the re-
mainder be denoted ?(103)
How do you place the numbers for

division? where the quotient? How is the operation performed?

quotient, and bring down another figure, and so on till the number shall contain the divisor.

PROOF.

107. Multiply the divisor by the quotient, (adding the remainder, if any) and, if it be right, the product will be equal to the dividend.

QUESTIONS FOR PRACTICE.

contained 235000 inhabitants, to the moon's distance from the what was the average quantity

of land to each person?

Ans. 24 acres.

20. The distance of the moon from the earth is 240000

earth?
Ans. 30.
21. Divide 17354 by 86.

Quot, 201. Rem. 68.

22. Divide 1044 by 9.

Quot. 116.

miles, and the diameter, or dis- | 23. Divide 34748748 by 24.

tance through the earth, is 8000 miles; how many diame

Quot. 1447864. Rem. 12.

24. 29702÷6-4950 Ans. ters of the earth will be equal 25. 279060-39865 Ans.

CONTRACTIONS OF DIVISION.

108. 1. Divide 867 doliars equally among 3 men, what will each receive?

Here we seek how many times 3 in 8, and finding Divis. 3) 867 Divid. it 2 times and 2 over, we write 2 under 8 for the first figure of the quotient, and suppose the 2,which 289 Quot. remains, to be joined to the 6, making 26. Then 3 in 26, 8 times, and 2 over. We write 8 for the next figure of the quotient, and place 2 before the 7, making 27, in which we find 3, 9 times. We therefore place 9 in the unit's place of the quotient, and the work is done. Division performed in this manner, without writing down the whole operation, is called Short Division.

I. When the divisor is a single figure;

RULE.-Perform the operation in the mind, according to the general rule, writing down only the quotient figures. 2. Divide 78904 by 4. 3. Divide 234567 by 9.

Quot. 19726.

Quot. 26063.

109. 4. Divide 238 dollars into 42 equal shares; how many dollars will there be in each?

42=6x7

7) 238-6 rem. 1st.

6) 33-3 rem. 2d.

If there were to be but 7 shares, we should divide by 7, and find the shares to be $33 each, with a remainder of 6 dollars; but as there are to be 6 times 7 shares, each share will be only one sixth of the above, or a little more than 5 dollars. In the example there are two remainders; the first, 6, is evidently 6 units of the given dividend, or 6 dollars; but the second, 3, is evidently units of the second dividend, which are 7 times as great as those of the first, or equal to 21 units of the first, and 21+6=27 dolls. the true remainder.

5 73+6=27 rem. Ans. 527 dolls.

II. When the divisor is a composite number. (90.)

RULE.-Divide first by one of the component parts, and that

What is Short Division?

What is the rule?

How do you multiply by a compo

site number?

Explain the operation.

quotient by another, and so on, if there be more than two, the last quotient will be the answer.

5. Divide 31046835 by 56-7

>8. Quo. 554407, Rem. 436. Divide 84874 by 48–6×8. Quo. 176818.

110. 7. Divide 45 apples equally among 10 children, how many will each child receive?

As it will take 10 apples to give each child 1, each child will evidently receive as many apples as there are 10's in the whole number; but all the figures of any number, taken together, may be regarded as tens, excepting that which is in the unit's place. The 4 then is the quotient, and the 5 is in the remainder; that is, 45 apples will give 10 children 4 apples and 5 tenths, or, each. And as all the figures of a number, higher than in the ten's place, may be considered hundreds, we may in like manner divide by 100, by cutting off two figures from the right of the dividend; and generally,

III. To divide by 10, 100, 1000, or 1 with any number of ciphers annexed;

RULE.-Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor; those on the left will be the quotient, and those on the right, the remainder.

8. Divide 46832101 by | mong 100 men, how much 10000. Quot. 4685-2. will each receive?

2101

9. Divide 1500 dollars a

111. 10. Divide 36556 into 3200 equal parts.

3200) 365156 (11 Quot.

32

45

32

1356 Rem.

Ans. 15 dolls.

Here 3200 is a composite number, whose component parts are 100 and 32; we therefore divide by 100, by cutting off the two right hand figures. We then divide the quotient, 365, by 32, and find the quotient to be 11, and remainder 13; but this remainder is 13 hundred, (109) and is restored to its proper place by bringing down the two figures which remained after dividing by 100, making the whole remainder, 1356. Hence

IV. To divide by any number whose right hand figures are ciphers;

RULE. Cut off the ciphers from the divisor, and as many figures from the right of the dividend; divide the remaining figures of the dividend by the remaining figures of the divisor, and bring down the figures cut off from the dividend to the right of the remainder.

What is the rule for multiplying by

1, with ciphers annexed? Give the reason for the operation.

How do you proceed when the divisor has ciphers in the right hand? Give the reason.

11. Divide 738064 by 2300. | 12. Divide 6095146 by 5600, Quot. 320, Rem. 2064. Quot. 10882848.

REVIEW,

27. If the multiplicand and mul 9. How would you find the diffe-tiplier were given, how would you rence between two numbers? (94.)