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18. Vermont contains 247 | distance through the earth, is townships, and is divided into 8000 miles; how many diame13 counties, what would be the ters of the earth will be equal average number of townships to the moon's distance from in each county? Ans. 19. the earth?

An3. 30. 19. Vermont contains 5640- 21. Divide 17354 by 86. 000 acres of land, and in 1820

Quot. 201. Rem. 68. contained 235000 inhabitants, 22. Divide 1044 by 9. what was the average quanti

Quot. 116. ty of land to each person? Ans. 24 acres.

23. Divide 34748748 by 24.

Quot. 1447864. Rem. 12. 20. The distance of the moon from the earth is 240000

24. 29702;0=49503 Ans. miles, and the diameter, or 25. 379060=398651 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 this 289 Quot.

first figure of the quotient, and suppose the 2, whicb 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 237 dollars into 42 equal shares; how many dollars will there be in each? 42=6X7

If there were to be kut 7 shares, we chould die 7)237.–6 rem. lst.

vide by 7, and find the shares to be $33 each, with

a remainder of 6 dollars; but as there are to be 6 6 33–3 rem. 2d.

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, 5 7X376=27 rem.

is evidently 6 units of the given dividend, or 6

dollars; but the seeond, 3, is evidently units. of Ans.

dolls.

522 the second dividend, which are 7 times as great 28 thiose of the fi.st, or equal to 21 units of the first, and 21+6=dollars, the true remainder.

X8. Quot.

554407, Rem. 43. 6. Divide 84874 by 48–6X8.

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

RULE.—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

Quot: 17687 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 the remainder; that is, 46 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 dividc by 10, 100, 1000, or 1 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 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. 4683 7.18: will each receive? 9. Divide 1500 dollars a

Ans. 15 dolls. 111. 10. Divide 36556 into 3200 equal parts.

Here 3200 is a composite number, whose com8200)365 56(11 Quot. ponent parts are 100 and 32; we therefore divide 82

by, 100, by cutting off the two right hand figures,

We then divide the quotient, 365, by 32, and find 45

the quotient to be 11, and remainder 13; but this 82

remainder is 13 hundred,[109], and is restored

to its proper place by bringing down the two fig1356 Rem. ures which remained after dividing by 100, make

ing 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 Sigures 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. Quot. 320, Rem. 2064.

Quot. 10884414.

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MISCELLANEOUS QUESTIONS. 1 If the minuend be 793, 12. How many ruds in a and the subtrahend be 598, piece of land 40 toda, long what is the remainder?

and 16 broad?

Ans. 195. Ans. 640. rods, or 4 acres., 2. If the minuend be 111, 13. The sum of two numand the remainder 63, what is bers is 75, and their difference the subtrahend? Ans. 48. is 15, what are the numbers?, 3. If the subtrahend be 645, the lessi 30+15=45, greater:

Ans. 75-15=60, 60-;230, and the remainder 131, what is the minuend? Ans. 776. 14. The difference of two

numbers is 723, and their sum 4. The sum of two num- is 1111, what are the numbers is 8392, and one of them

bers?

194 is 4785, what is the other?

Ans. Ans. 3607. 5. The least of two num

. 15. If a man travel 35 miles bers is 77, and their difference in 6 weeks and 3 days, al

a day, how far will he travel is 99, what is the greater?

Ans. 176.
lowing 6 days to a week?

Ans, 1365 miles. 6. A certain dividend is

16. What sum of money 2340, and the quotient is 156, must be divided among 18 what is the divisor? Ans. 15.

men so as to give each man 7. If the divisor be 32, and $112?

Ans. $2016. the quotient 204, what is the

17. A man raised 64562 dividend? Ans, 6528.

bushels of corn on 1565 acres, 8. A certain product is how many bushels was that 484848, and the multiplicand to the acre? Ans. 41. is 1036, what is the multipli

18. If I plant in 14 rows er?

Ans. 468.

2072 fruit trees, and set the 9. If a person spend 8 cts. trees 25 feet asunder, how a day, how much will he many feet long are the rows? spend in a year, or 365 days?

Ans. 3075 feet. Ans. 2920 cts.$29.20.

19. Subtract 30079 out of 10. How many square feet ninety-three millions as often in a piece of ground 17 feet as it can be done, and say long, 13 ft. wide?(36, 61) how much the last remainder Ans. 221 feet. exceeds or falls short of

21180? 11. If a floor containing

Ans. 4631 exceeds, 242 feet be 22 feet long, how vide is it? Ans. 11 feet.

REVIEW. 112. 1. What are the funda- 19. By what name would you mental operations in this section? call the nuinber divided?[105)

Ans. Addition and Subtraction. 20. What would you call the

2. What relation have Maltipli- other number? cation and Division to these? (83, 21. By what name would you 101)

call the result of the operation? 3. When two or more numbers 22. Where there is a part of the are given, how do you find their dividend left after perforining the sim?

operation, what is it called? 4. What is the method of per- 23. How can you denote the diforming the operation?(81)

vision of this remainder ?[103] 5. When the given numbers are 24. If the divisor and dividend all equal, what shorter method is were given, how would you find the there of finding their sum?(83) quotient? 6. How is Multiplication per

25. If the dividend and quocient formed?(88)

were given, how would you fod the 7. What are the given numbers divisor? employed in Multiplication called? 26. If the divisor and quotient (87)

were given, how would you find tha. 8. What is the result of the ope- dividend? ration called?(87)

27. If the multiplicand and mul9. How would you find the diffe- tiplier were given, how would you rence between two numbers?(94) find the product? 10. By what names would you

28. If the multiplicand and procall the two numbers?(98)

duct were given, how would you 11. What is the difference called? | find the multiplier?

12. If the minuend and subtra- 29. If the multiplier and product hend were given, how would you were given, how would you find the find the remainder?

multiplicand? 13. If the minuend and remain- 30. When the price of an article der were given, how would you find is given, how do you find the price the subtrahend?

of a number of articles of the same 14. If the subtrahend and remain-kind?[83] der were given, how would you find 31. Does the proof of an ariththe minuend?

metical operation demonstrate its 15. If the sum of two numbers, cowreetness![82] What then is its aud one of them were given, how use? would you find the other?

* 16. If the greater of two num- NOTE,—The definitions of such bers and their difference be given, of the following terms as have not how would you find the less? been already explained, may be

17. If the less of two numbers found in a dictionary. and their difference be given, how What is Arithmetic? What is a would you find the greater?

Science? Number? Notation? Nu 18. How would you find how meration? Quantity? Question? many times one number is contain. Rule? Answer? Proof? Principle? ed in another?

Illustration? Explanation ?

DECIMALS AND FEDERAL MONEY.

DECIMALS. 113. The method of forming numbers, and of expressing them by figures, has been fully explained in the articles on Numeration. (71, 72, 73) But it frequently happens that we bave occasion to express quantities, which are less than the one fixed upon for udity. Should we make the foot, for in stance, our unit measure, we should often have occasion to express distances which are parts of a foot. This has ordinarily been done by dividing the foot into 12 equal parts, called inches, and each of these again into 3 equal parts, called barley corns. (38) But divisions of this nature, which are not conformable to the general law of Notation, (73) necessarily embarrass calculations, and also encumber books and the memories of pupils, with a great number of irregular and perplexing tables. Now, if the foot, instead of being divided into 12 parts, be divided into 10 parts, or tenths of a foot, and. each of these again into 10 parts, which would be tenths of tenths or hundredths of a foot, and so on to any extent found necessary, making the parts 10 times smaller at each division ;-then in recomposing the larger divisions from the smaller, 10 of the smaller would be required to make one of the next larger, and so on, precisely as in whole numbers. Hence, figures expressing tenths, hundredths, thousandths, &c. may be written towards the right from the place of units, in the same manner that tens, hundreds, thousands, &c. are ranged towards the left; and as the law of increase to wards the left, and of decrease towards the right, is the same, those figures which express parts of a unit may obviously be managed precisely in the same manner as those which denote integers, or whole numbers. But to prevent confusion, it is customary to separate the figures expressing parts from the integers by a point, called a separatrir. The points used for this purpose are the period and the comma, the former of which is adopted in this work; thus to express 12 feet and 3 tenths of a foot, we write 12.3 ft. for 8 feet and 46 hundredths, 8.46 feet.

DEFINITIONS. 114. Numbers which diminish in value, from the place of units towards the right hand, in a ten fold proportion, (as

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