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Norf..- after bringing down a figure to the remainder, i: 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 eine 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 re mainder, if any) and, if it be right, the product will be equal to the dividend.


6, If 30114 dollars be die 11. What number must I vided equally among 63 men, multiply by 25, that the prohow many

dollars will each duct may be 625? Ans. 25. one receive?

12. If a certain number of 63)30114(478 dolls. Ans. 252

men, by paying 33 dollars each, paid 726 dollars, what

was the number of men? 491

Ans. 22. 441

13. The pols in a certain 504

town pay 750 dollars, and the

number of polls is 375, what 504

does each poll pay?

Ans. 2 dolls. n. If a man's income be

14. If 15 horses were sold 1460 dollars a year, how much in the West Indies for 9900 is that a day? Ans. 4 dolls.

dollars, what was the average 8. A man dies leaving an price of each? Ans. $220. Estate of 7875 dollars to his 7

15. An army of 97440 men sons, what is each son's share? Ans. 1125 dolls.

was divided into 14 equal di

visions, how many men were 3. A field of 34 acres pro- there in each? Ans. 6960. duced 1020 bushels of

corn, how much was that per acre?

16. A gentleman,who own

ed 520 acres of land, purAns. 30 bush. chased 376 acres more, and 10. A privateer of 175 men then divided the whole into took a prize worth 20650 dol- eight equal farms; what was lars, of which the owner of the size of each? the privateer had one half,

Ans. 112 acres and the rest was divided e- 17, A certain township qually among the men; what contains 30000 acres, how was oach man's share?

many lots of 125 acres each Ane. 59 dolls. does it contain? Ans. 240.

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 carth?

Anz, 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. ity 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=4950} Ans. miles, and the diameter, or

25. 379960=398655 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, which

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 isæsingle 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 I. Quot. 19726.

Quot. 26063. 109. 4. Divide 237 dollars into 42 equal shares; how many dollars will there be in each? 4256X17

If there were to be kut:7 shares, we chould di-7)237-6 rem. lst.

vide by 7, and find the shares to be $33 cach, urith
a remainder of 6 dollars; bat as there are to be 6

times:7 shares, each share will be only one sixth 6)333 rem. 2d.

of the above, or a little more than 5 dollars. In

the example there are two remainders; the first, 6, 5

-6 is evidently 6 units of the given dividend, or 17X376=27 rom.

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

the second dividend, which are 7 times as great as Niese of the first, or equal to 21 waits of the first, and 21,44 dollars, she lerne remainde

Ans. 542

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II. When the livisor 23 ( coinposite number.(90)

Rure.-Divide first hy 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 31045835 by 5li=76. Divide 84874 by 48=6X8.

Quot: 176818. 110. 7. Divide 45 apples equally among 10 children, how many wia each child receive?

As it will take 10 apples to give each child 1, each child will evidently receive as many appiles as there are 10's in the whole number'; but all the figures of any number, tisken lugether, may be regarded as tens, excepting that which is in the unit's place. 'l'he 4 then is the quotient, and the õ is the remainiler; 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 considereil lundreds, we may in like manner divide by 100, by culling off two figures from the right of the dividend; and, generally,

II). 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 diviscr; chose on the left will be the quotient, and those on the right, the remainder.

8. Divide 4633-191 by, mong lo men, how much 10000, Quot. 46532. will cach receive? 9. Divide 1500 dollars a

Ans. 15 dolls. 111. 10. Divide 35556 into 3200 cqual parts.

Here 3200 is ? cuiposiie number, wlose com$2 00)365 56(11 Quot. ponent parts are 100 and 32; we therefore divide

by 100, by cuing off the two right hand figures.

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

the quotient to be 11, and remainder 13; but this 32

remainder is 13 hundredl, [109), and is restored

to its proper place by biinging down the two fig1356 Rom. ucs, which remained after dividing by 100, mak.

ing the whicle remainder, 1336. Hence, IV. To divide by any number whose right hand figures are ciphers :

RULE.-Cut off the ciphers from the divisor, and as many Sgures from the right of the dividend; divide the remaining tigures 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 rernainder. Ji Nivide 738064 hy 2300. 12. Divide 6095146 by 5600.



MISCELLANEOUS QUESTIONS. 1 If the minuend be 793, 12. How many ruds in a and the subtrahend be 598, picce of land 40 rods 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?

Ans. 75—15=50, 60; 2=30, 3. If the subtrahend be 645,

the less. 30+15=45, greater. 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 num'bers is 8392, and one of them


194 is 4785, what is the other?

Ans. Ans, 3607.

15. If a man travel 35 miles 5. The least of two numbers i3 77, and their difference a day, how far will he travel

in 6 weeks and 3 days, alis 99, what is the greater?

lowing 6 days to a week? Ans. 176,

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. trces 25 feet asunder, how a day, how much will be many feet long are the rows? spend in a year, or 365 days?

Ans. 3675 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 Jong, 13 ft. wide?(36, 61) how much the last remainder

Ans. 221 fcet. exceeds or falls short of 11. If a floor containing

21180? 242 feet be 22 feet long, how

Ans. 4631 exceeds. wide is it? Ans. Il feet.

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

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

2. What relation have Multipli- other number? cation and Division tu thesc? (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 performing the sun?

operation, what is it called? 4. What is the method of


23. How can you denote the die forming 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 quotiem formed?(88)

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

were given, how would you fiod the 8. What is the result of the


dividend? ration called?(87)

27. If the multiplicand and male. 9. 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 pro. call 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 artiole: der were given, how would you find | is given, how do you find the prioe 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 arith, the minuend?

metical operation demonstrate ito 15. If the sum of two numbers, correctness?[82] What theo is ite 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 if a would you find the greater?

Science? Number? Notation? Now 18. How would you find how meration? Quantity? Question? many times one number is contain- Rule? Answer? Proof? Principles ed, in apopber?

Illustration? Explanation

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