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107.

DIVISION.

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

6. If 30114 dollars be divided equally among 63 men, how many dollars will each one receive?

63)30114(478 dolls. Ans.
252

491

441

504 504

7. If a man's income be 1460 dollars a year, how much is that a day? Ans. 4 dolls.

8. A man dies leaving an estate of 7875 dollars to his 7 sons, what is each son's share? Ans. 1125 dolls.

9. A field of 34 acres produced 1020 bushels of corn, how much was that per acre?

Ans. 30 bush.

10. A privateer of 175 men took a prize worth 20650 dollars, of which the owner of the privateer had one half, and the rest was divided equally among the men; what was each man's share?

Ans. 59 dolls.

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14. If 45 horses were sold in the West Indies for 9900 dollars, what was the average price of each? Ans. $220.

15. An army of 97440 men was divided into 14 equal divisions, how many men were there in each? Ans. 6960.

16. A gentleman, who owned 520 acres of land, purchased 376 acres more, and then divided the whole into eight equal farms; what was the size of each?

Ans. 112 acres. 17. A certain township contains 30000 acres, how many lots of 125 acres each does it contain? Ans. 240.

18. Vermont contains 247 | townships, and is divided into 13 counties, what would be the average number of townships in each county? Ans. 19.

19. Vermont contains 5640000 acres of land, and in 1820 contained 235000 inhabitants, what was the average quantity of land to each person?

Ans. 24 acres.

20. The distance of the moon from the earth is 240000 miles, and the diameter, or

distance through the earth, is
8000 miles; how many diame-
ters of the earth will be equal
to the moon's distance from
the earth?
Anз. 30.

21. Divide 17354 by 86.
Quot. 201. Rem, 68.
22. Divide 1044 by 9.
Quot. 116.

23. Divide 34748748 by 24. Quot. 1447864. Rem. 12. 24. 29702÷6=4950} Ans.

25. 279960=398651⁄2 Ans.

'CONTRACTIONS OF DIVISION.

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

Divis. 3) 867 Divid.

Here we seek how many times 3 in 8, and finding it 2 times and 2 over, we write 2 under 8 for the 289 Quot. first figure of the quotient, and suppose the 2, which remains, to be joined to the 6, making 26. Then 8 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.

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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.

Quot. 19726.

3. Divide 234567 by 9.
Quot. 26063.

109. 4. Divide 237 dollars into 42 equal shares; how many dollars will there be in each?

42=6X7 7)237-6 rem. 1st.

6)33-3 rem. 2d.

In

If there were to be kut 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. 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 2146dollars, the Larue remainder.

5

7X3+6=27 rem. Ans. 5230 dolls.

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

ROLE.-Divide first by one of the component parts, and that 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 6. Divide 84874 by 48-6X8. X8. Quot. 554407, Rem. 43. Quot: 176818.

110. 7. Divide 45 apples equally among 10 children, how many wid 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 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 I with any number of eiphers annexed:

RULE. Cut off as many figures from the right hand of the dividend as there are ciphers in the diviser; those on the left will be the quotient, and those on the right, the remainder. 8. Divide 46832191 by | mong 100 men, how much 10000. Quot. 4653,21 will each receive?

9. Divide 1500 dollars a

111. 10. Divide 36556 into 3200 equal parts.

82

45 32

Ans. 15 dolls.

Here 3200 is 2. composite number, whose com

82 00)365 56(11 Quot. ponent parts are 100 and 32; we therefore divide by 100, by cuting 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 binging 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:

1356 Rem.

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 divi sor, and bring down the figures cut off from the dividend to the right of the remainder.

11 Divide 738064 by 2300.

12. Divide 6095146 by 5600.

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MISCELLANEOUS QUESTIONS.

1 If the minuend be 793, and the subtrahend be 598, what is the remainder?

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6. A certain dividend is

2340, and the quotient is 156, what is the divisor? Ans. 15.

7. If the divisor be 32, and the quotient 204, what is the dividend? Ans. 6528.

8. A certain product is 484848, and the multiplicand is 1036, what is the multiplier? Ans. 468.

9. If a person spend 8 cts. a day, how much will he spend in a year, or 365 days?

Ans. 2920 cts.=$29.20.

10. How many square feet in a piece of ground 17 feet long, 13 ft. wide?(36, 61)

Ans. 221 feet. 11. If a floor containing 242 feet be 22 feet long, how wide is it? Ans. 11 feet.

12. How many rods in a piece of land 40 rods long and 16 broad?

Ans. 640 rods, or 4 acres.

13. The sum of two numbers is 75, and their difference is 15, what are the numbers? the less. 30+15-45, greater. Ans. 75—15—60, 60÷2—30,

14. The difference of two numbers is 723, and their sum is 1111, what are the numbers?

194

917 Ans.

15. If a man travel 35 miles

a day, how far will he travel in 6 weeks and 3 days, allowing 6 days to a week?

Ans. 1365 miles.

16. What sum of money must be divided among 18 men so as to give each man $112? Ans. $2016.

17. A man raised 64562 bushels of corn on 1565 acres, how many bushels was that to the acre? Ans. 41.

18. If I plant in 14 rows 2072 fruit trees, and set the trees 25 feet asunder, how many feet long are the rows?

Ans. 3675 feet.

19. Subtract 30079 out of ninety-three millions as often as it can be done, and say how much the last remainder exceeds or falls short of 21180?

Ans. 4631 exceeds.

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112. 1. What are the fundamental operations in this section?

Ans. Addition and Subtraction. 2. What relation have Multiplication and Division to these? (83, 101)

3. When two or more numbers are given, how do you find their

sun?

4. What is the method of performing the operation?(81)

5. When the given numbers are all equal, what shorter method is there of finding their sum? (83)

6. How is Multiplication performed?(88)

7. What are the given numbers employed in Multiplication called? (87)

8. What is the result of the operation called?(87)

9. How would you find the difference between two numbers?(94) 10. By what names would you call the two numbers?(98)

11. What is the difference called? 12. If the minuend and subtrahend were given, how would you find the remainder?

13. If the minuend and remainder were given, how would you find the subtrahend?

14 If the subtrahend and remainder were given, how would you find the minuend?

15. If the sum of two numbers, aud one of them were given, how would you find the other?

16. If the greater of two numbera and their difference be given, how would you find the less?

17. If the less of two numbers and their difference be given, how would you find the greater?

18. How would you find how many times one number is contained, in another?

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19. By what name would you call the number divided [105] 20. What would you call the other number?

21. By what name would you call the result of the operation?

22. Where there is a part of the dividend left after performing the operation, what is it called?

23. How can you denote the division of this remainder?[103]

24. If the divisor and dividend were given, how would you find the quotient?

25. If the dividend and quotin were given, how would you find the divisor?

26. If the divisor and quotient were given, how would you find the dividend?

27. If the multiplicand and multiplier were given, how would you find the product?

28. If the multiplicand and pro duct were given, how would you find the multiplier?

29. If the multiplier and product were given, how would you find the multiplicand?

30. When the price of an article is given, how do you find the prioe of a number of articles of the same kind?[83]

31. Does the proof of an arith-. metical operation demonstrate its correctness?[82] What then is its use?

NOTE. The definitions of such of the following terms as have not been already explained, may be found in a dictionary.

What is Arithmetic? What is a Science? Number? Notation? Nomeration? Quantity? Question? Rule? Answer? Proof? Principle? Illustration? Explanation!

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