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78. When the divisor consists of two or more


To illustrate more clearly the method of operation, we will first take an example usually performed by Short Division.

1. How many times is 8 contained in 2528?


Divisor. Divid'd. Quotient.

8 ) 2528 ( 316






down the 2 tens of the dividend, and consider the 12 tens as a second partial dividend. Then, 8 is contained in 12 tens 1 ten time and a remainder; 8 multiplied by 1 ten produces 8 tens, which, subtracted from 12 tens, leave 4 tens. To this remainder we bring down the 8 units, and consider the 48 units as the third partial dividend. Then, 8 is contained in 48 units 6 units times. Multiplying and subtracting as before, we find that nothing remains, and the entire quotient is 316.


Divisor. Divid'd. Quotient.

23) 4807 (209


SOLUTION. As 8 is not contained in 2 thousands, we take 2 and 5 as one number, and consider how many times 8 is cortained in this partial dividend, 25 hundreds, and find that it is contained 3 hundreds times, and a remainder. To find this remainder, we multiply the divisor, 8, by the quotient figure, 3 hundreds, and subtract the product, 24 hundreds, from the partial dividend, 25 hundreds, and there remains 1 hundred. To this remainder we bring

2. How many times is 23 contained in 4807 ?

SOLUTION. We first find how many times 23 is contained in 48, the first partial dividend, and place the result in the quotient on the right of the dividend. We then multiply the divisor, 23, by the quotient figure, 2, and subtract the product, 46, from the part of the dividend used, and to the remainder bring down the next figure of the dividend, which is 0, making 20, for the second partial dividend. Then, since 23 is contained in 20 no times, we place a cipher in the quotient, and bring down the next figure of the dividend, making a third partial dividend, 207; 23 is contained in 207, 9 times; multiplying and subtracting as before, nothing remains, and the entire quotient is 209.



79. When the process of division is performed mentally, and the results only are written, the operation is termed Short Division.

When the whole process of division is written, the operation is termed Long Division.

Short Division is generally used when the divisor is a number that will allow the division to be performed mentally.

From the preceding illustrations we derive the following general rule:

RULE.-I. Write the divisor at the left of the dividend, as in short division.

II. Divide the least number of the left hand figures in the dividend that will contain the divisor one or more times, and place the quotient at the right of the dividend, with a line between them.

III. Multiply the divisor by this quotient figure, subtract the product from the partial dividend used, and to the remainder bring down the next figure of the dividend.

IV. Divide as before, until all the figures of the dividend have been brought down and divided.

V. If any partial dividend will not contain the divisor, place a cipher in the quotient, and bring down the next figure of the dividend, and divide as before.

VI. If there is a remainder after dividing all the figures of the dividend, it must be written in the quotient, with the divisor underneath.

1. If any remainder is equal to, or greater than the divisor, the quotient figure is too small, and must be increased.

2. If the product of the divisor by the quotient figure is greater than the partial dividend, the quotient figure is too large, and must be diminished.

PROOF.-1. The same as in short division. Or,

2. Subtract the remainder, if any, from the dividend, and divide the difference by the quotient; if the result is the same as the given divisor, the work is correct.

80. The operations in long division consist of five principal steps, viz.:

1. Write down the numbers.

2. Find how many times the divisor is contained in the partial dividend.

3. Multiply.

4. Subtract.

5. Bring down another figure.

3. Find how many times 36 is contained in 11798.




Divisor. 36)11798 (327 Quotient.








82)89634 (1093




327 Quotient. 36 Divisor.

26 Remainder.

26 Remainder.

11798 Dividend.

4. Find how many times 82 is contained in 89634.






89634 Dividend.
8 Remainder.

Quotient. 1093)89626 (82 Divisor.





5. Find how many times 154 is contained in 32740.

6. Divide 32572 by 34.

7. Divide 1554768 by 216.

Ans. 958.

Ans. 7198.

8. Divide 5497800 by 175. 9. Divide 3931476 by 556. 10. Divide 10983588 by 132. 11. Divide 73484248 by 19. 12. Divide 8121918 by 21. 13. Divide 10557312 by 16. 14. Divide 93840 by 63. 15. Divide 352417 by 29. 16. Divide 51846734 by 102. 17. Divide 1457924651 by 1204. 18. Divide 729386 by 731. 19. Divide 4843167 by 3605. 20. Divide 49816657 by 9101. 21. Divide 75867308 by 10115.

Ans. 31416.

Ans. 7071.

Ans. 83209.

Ans. 3867592.

Ans. 386758.

Ans. 659832.


Rem. 33.

Rem. 9.

Rem. 32.

Rem. 1051.

Rem. 579.

Rem. 1652.

Rem. 6884.
Rem. 4808.

22. Divide 28101418481 by 1107.
23. Divide 65358547823 by 2789.
24. Divide 102030405060 by 123456.
25. Divide 48659910 by 54001.

Quotients. Rem. 25385201. 974. 23434402. 645. 826451. 70404. 901. 5009.



26. Divide 2331883961 by 6739549. 27. A railroad cost one million eight hundred fifty thousand four hundred dollars, and was divided into eighteen thousand five hundred and four shares. What was the value of each share? Ans. $100.

28. If a tax of seventy-two million three hundred twenty thousand sixty dollars is equally assessed on ten thousand seven hundred thirty-five towns, what amount of tax must each town pay? Ans. $6736,2100. 29. If 213 college libraries contain 942321 volumes, what is the average number of volumes to each library? Ans. 4424 volumes. 30. A man bought 240 acres of land at $15 an acre, giving in payment horses valued at $180 apiece. How many horses did he give?




81. When the divisor is a composite number.

1. If 3270 dollars are divided equally among 30 men, how many dollars will each receive?

SOLUTION. If 3270 dollars are divided equally among 30 men, each man will receive as many dollars as 30 is contained times in 3270 dollars. 30 may be resolved into the factors 5 and 6; and we may suppose the 30 men divided into 5 groups of 6 men each; dividing the 3270 dollars by 5, the number of groups, we have 654, the number of dollars to be given to each group; and dividing the 654 dollars by 6, the number of men in each group, we have 109, the number of dollars that each man will receive.


5 3270

6 654
109 Ans,

Hence we have the following rule:


RULE. - Divide the dividend by one of the factors, and the quotient thus obtained by another, and so on if there are more than two factors, until every factor has been made a divisor. The last quotient will be the quotient required.

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