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Had this remainder been 3 dollars, it is evident that each man would have had 3 times of a dollar, that is of a dollar, more ; and in all cases where there is a remainder, we may obtain the true quotient by placing the divisor unider the remaindet, with a line between, as above.

Remainder 1 Thus

shows that the divisor 4 is contained Divisor 4 in the remainder one fourth of a time.

19. How many cwt. of rice, at 4 dollars a cwt., may be bought for 947 dollars ?

Ans. 236cwt. 20. How many cwt. of sugar, at 9 dollars per cwt., can be bought for 2944 dollars ?

Ans. 3275 21. How many barrels of pork, at 11 dollars a barrel, can be bought for 2478 dollars ? Ans. 22511 barrels.

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LONG DIVISION. When the divisor exceeds 12, we cannot conveniently perform the operation in the mind; we therefore set the quotient figures on the right hand of the dividend, and write down the whole computation at full length, and this is called long division.

RULE.

I. Find how many times the divisor is contained in the least number of the left hand figures of the dividend, that will contain it once, or more: place the figure expressing the number of times, to the right hand of the dividend for the first quotient figure.

II. Multiply the divisor by this quotient figure, and place the product under that part of the dividend used, and subtract it therefrom.

III. Bring down the next figure of the dividend to the right hand of the remainder, and divide this number as before. Thus proceed till you have brought down, and divided, all the figures of the dividend.

Note. 1. If the product of the divisor by any quotient figure, be greater than that part of the dividend used, it shows that the quotient figure is too large, and must be diminished.-If the remainder at any time be equal to

or

greater than, the divisor, the quotient figure is too small; and must be increased.

2. When you have brought down a figure to the right hand of the remainder, if the number made up be less than the divisor, you must place a cipher in the quotient, and bring down the next figure of the dividend.

Proof.-The method of proof is the same as in Short Division.

EXAMPLES. 1. How many times is 13 contained in 2785 ? Operation.

We find that the divisor Divisor. Dividend. Quotient. 1 3 ) 2 7 8 5 (2 14

(13) is contained in the two 2 6

first figures of the dividend,

2 times. We place the fig 18

ure (2) at the right hand of

the dividend for the first quo1 3

tient figure, and multiply the

divisor (13) by it, and place 5 5 5 2

the product (26) under the

two first figures of the diviRemainder 3

dend, and subtract it there

from, and I remains. To the Quotient 214

right hand of this remainder Divisor X13

bring down (8) the next fig

ure of the dividend, which 6 4 2

makes 18. We find that 13 2 1 4

is contained in 18, 1 time;

placing 1 in the quotient, and 27 82

13 under the 18, subtract it Remainder +3

therefrom, and 5 remain, to the

right hand of which, we bring 2 7 8 5=Divid. down (5) the next figure of the dividend, which makes 55. We then find that 13 is contained in 55, 4 times. Placing 4 in the quotient, multiply the divisor (13) by it, and place the product (52) under the 55, subtract it therefrom, and the remainder is 3. Thus we have brought down and divided all the figures of the dividend. Hence we find that 13 is contained in 2785, 214 limes, and 3 remains.

Proof.

2. Divide $2448 equally among 16 men. Divis. Divid. Quot. 16 ) 2 4 4 8 ( 1 5 3

16

84
80

48

4 8 Rem.

3. How many times is 21 contained in 2859 ?

Divis. Divid. Quot.
2 1 ) 2 8 5 9 (1 3 6 i

2 1

7 5
6 3
1 29

1 2 6
Rem. 3
Divisor 21

5. How many times is 24
contained in 8760 ?
Divis. Divid. Quot.
2 4 ) 8760 (3 6 5 Ans:

72
1 5 6
1 4 4

1 2 0
1 2 0

4. Divide 3625 by 18. Divis. Divid. Quot. 18 ) 3 6 2 5 ( 2 0 1

3 6

2 5

18 Rem.

6. How many times is 13 contained in 29877 ?

Ans. 229815 times. 7. How many times can you have 16 in 5520 ?'

Ans. 345 times. 8. If a man travel 630 miles in 15 days, how many miles is that a day?

Ans. 42 miles. 9. What is the quotient of 55081 divided by 17 ?

Ans. 324014: 10. 25 workmen received $3150, to be equally divided among them ; how many dollars did each one receive ?

Ans. $126. 11. If the divisor be 29, and the dividend 72740, what is the quotient?

Ans. 250879 12. How often does 106215 contain 365 ? Ans. 291. 13. Divide 6832 by 16.

Ans. 427. 14. Divide 28656 by 36.

Ans, 796. 15. Divide 37088 by 61.

Ans. 608. 16. Divide 410642 by 79.

Ans. 5198

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When there are ciphers at the right hand of the divisor.

RULE.

Cut off the ciphers in the divisor, and the same number of figures from the right hand of the dividend ; then divide the remaining ones as usual, and to the right hand of the remainder, (if any) place the figures which were cut off from the dividend, and you will have the true remainder.

EXAMPLES.

1. Divide 7380964 by 23100. 2. Divide 8978485 by 8000.

Divis. Divid. Quot. Divis. Divid. 231'00)73809'64(31923166 8'000)8978'485 693

Quot. 1122-2485 trüe rem. 450 231

3. Divide 741625 by 900. 2199

9'00)7416'25 2079

Quot. 824-25 Rem. 12064 4. Divide 75694 by 2500.

Ans. 30,694 5. Divide 1255000 dollars equally among 5000 men, and how many dollars will each one have ? Ans. 251.

6. Divide 49184145 by 3600. Ans. 136623% 08

7. How many times is 49000 contained in 421400000 ?

Ans. 8600. 8. Divide 3259450000 by 60000. Ans. 543241000% 9. How many times does 11659112 contain 89000 ?

Ans. 131370

112

10. Divide 436250000 by 125000.

Ans. 3490.

MORE EXAMPLES.

1. Divide 149596478 by 120000. 2. Divide 654347230 by 901000. 3. Divide 457878695 by 9736000.

Rem. 76478.
Rem. 221230.
Rem. 286695.

CASE II.

When the divisor is a composite number ; that is, when it is the product of any two numbers in the Table.

RULE. Divide the dividend by one of the component parts, or factors, and the quotient thence arising by the other, the last quotient will be the answer.

Note. The true remainder is found by multiplying the last remainder by the first divisor, and adding in the first remainder.

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