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quotient figure underneath, as before. Proceed in this way until every part of the dividend is thus divided, and the result will be the quotient sought.

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Quotients. 33946768 105596437 141973831

24691357 111946492 23762387 20166474 17964186 33081425 13698246

67893536 by 2 316789311 by 3 567895326 by 123456789 by 671678953 by 166336711 by 161331793 by 161677678 by 363895678 by 164378956 by 12

78950077 by 678956671 by 667788976 by 5 777777777 by 888888888 by 7 999999999 by 8 100000000 by 7

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* From this and subsequent examples it will be seen that fractions

se from division, and are parts of a unit; that the denominator of the fraction represents the divisor, and shows into how many parts the given number or quantity is divided, and the numerator, being the remainder, shows how many units of the given quantity or dividend remain undivided. By writing the numerator over the denominator in the form of a fraction, we signify that it is to be divided by the denominator; and when placed at the right hand of the whole number in the quotient, the fraction becomes a part of the quotient, and, as such, is as much less than a unit, as the numerator is less than the denominator.

LONG DIVISION.

CASE I.

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EXAMPLE. 1. A prize, valued at $3978, is to be equally divided among 17 men. What is the share of each ?

The object of this quesOPERATION.

tion is to find how many Dividend.

times 3978 will contain Divisor. 17) 3978 ( 234 Quotient. 17. or how many times 34 17

must 17 be subtracted 57 1638

from 3978, until nothing 51 234

shall remain. We first 68 3978 Proof. inquire, how many times

the first two figures of 00 Remainder.

the dividend will contain

the divisor; that is, how many times 39 will contain 17. Having found it to be 2 times, we write 2 in the quotient and multiply the divisor, 17, by it, and place their product 34 under 39, from which we subtract it, and find the remainder to be 5, to which we annex the next figure of the dividend, 7. And having found that 57 will contain the divisor 3 times, we write 3 in the quotient, multiply it by 17, and place the product 51 under 57, from which we subtract it, and to the remainder, 6, we annex the next figure of the dividend, 8, and inquire how many times 68 will contain the divisor, and find it to be 4 times. And having placed the product of 4 times 17 under 68, we find there is no remainder, and that 3978 will contain 17, the divisor, 234 times; that is, each man will receive 234 dollars. To prove our work is right, we reason thus. If one man receives 234 dollars, 17 men will receive 17 times as much, and 17 times 234 are 3978, the same as the dividend ; and this operation is effected by multiplying the divisor by the quotient. The student will now see the propriety of the following

RULE Place the divisor and dividend as under the preceding rule, and draw a curved or perpendicular line on the right of the dividend. Then observe how many figures of the dividend, counting from left to right, must be taken to contain the divisor one or more times, but never exceeding nine times, and ascertain how many times these figures will contain the divisor, placing the quotient figure on the right hand of the dividend. Then multiply the divisor by this quotient figure, and place the product in order under the figures of the dividend that were taken. Subtract this product from the part of the dividend above it, and to the difference bring down and annex the next figure of the dividend. Divide this number by the divisor, and place the quotient figure on the right of the one already found. Multiply the divisor by the quotient figure last found, and subtract the product from the number last divided, and bring down and annex as before, till the last figure of the dividend is taken ; and the several figures on the right of the dividend will be the quotient required. The difference between the number last divided and the last product will be the remainder, which, with the divisor, will form a fraction, as under the preceding rule.

Note 1. - It will often happen, that, after a figure is brought down and annexed to a remainder, the number will not contain a divisor. In such a case, a cipher is to be placed in the quotient, and the next figure to be brought down and annexed, and thus till the number formed shall be large enough to contain the divisor. Sometimes it will be necessary thus to place several ciphers in succession in the quotient.

Note 2. — The proper remainder is in all cases less than the divisor; and if, at any time, the subtraction named in the rule gives a remainder larger than the divisor, we discover at once, that an error has been com. mitted in the division, and that the quotient figure must be increased.

PROOF. Division may be proved by Multiplication, by Addition, by casting out the 9's, or by Division.

By the first method, we multiply the quotient by the divisor, adding to the product the remainder, and the result, if the work be right, is equal to the dividend.

By the second method, we add up the several products of the several quotient figures by the divisor, together with the remainder, and the result, if the work is right, is like the dividend. See Example 2.

To prove Division by casting out the 9's, we find the excess of 9's in the divisor and also in the quotient, and multiply these excesses together and find the excess in their product. We then subtract the remainder from the dividend, and find the excess of 9's in the difference, which, if the work is right, will be equal to the excess found in the product of the excesses above named. See Example 3.

To prove Division by Division itself, we subtract the remain. der from the dividend, and divide the difference by the quotient, and, if the work is right, the result will be equal to the original divisor. See Example 4.

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Rem.

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5. Divide 6. Divide 7. Divide 8. Divide

9. Divide 10. Divide 11. Divide 12. Divide 13. Divide 14. Divide 15. Divide 16. Divide 17. Divide 18. Divide 19. Divide 20. Divide 21. Divide 22. Divide 23. Divide

6756785 by

789636 by 7967848 by 16785675 by

675753 by 5678910 by 6716394 by 1167861 by 7861783 by 1678567 by 87635163 by 34567890 by 78911007 by 78963167 by 671616589 by 471361876 by 300700801 by

10000000 by 199999999 by

Quotients.
193051

17166
153227
275175

17327 69255 71451

8650 90365

4598
226447

5091
2149

1728
10924
12812
29959
1000

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24. Divide

6716789513 by 7816789 2167762 25. Divide

1613716131 by 3151638 77475 26. Divide 121932631112635269 by 123456789 27. Divide 213255467083 by 30204

4267 28. Divide 4814703652065 by 800070

2355 29. Divide 2650726050934 by 530071

1234 30. Divide 395020613 by 4444

2341 31. Divide 9594004321 by 78000

4321 32. Divide 162068563389 by 40506

6789 33. Divide 5427563776896 by 808070

7896 34. Divide 475065610503 by 481007

8967 35. Divide 8794170278 by 470

278 36. Divide 70006876 by 7000

6876 37. Divide 204060808062747 by 2020202

2345 38. Divide 700003456 by 10000

3456 39. Divide 7207276639 by 9009

4567 40. Divide 63126068678 by 70070

5678 41. Divide 3394240208391 by 706007

6785 42. Divide 169233137936 by 10080

7856 43. Divide 915527086796874 by 3011101

8567 44. Divide 454115186870257 by 500123

8765 45. Divide 12032109124169380 by 6007023

1357 CASE II. To divide by any number with ciphers annexed. Cut off the ciphers from the divisor, and the same number of figures from the right hand of the dividend. Then divide the remaining fig. ures of the dividend by the remaining figures of the divisor, and the result will be the quotient. To complete the work, annex to the last ren mainder found by the operation the figures cut off from the dividend, and the whole will form the true remainder.

EXAMPLE. 1. Divide 36378967 by 31000.

31,000)36378,967(1173

31
. 53
31
227

217
108

93
15967 Remainder.

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