REM. 1.- The same result is obtained by short division, by putting the first figure of the quotient under the left-hand figure of the dividend (when it is contained in it), as it is of the same order.

REM. 2.-If the unit of the divisor is not contained in the first unit of the dividend, then the first figure of the quotient will be of the same order as the second figure of the dividend and should be placed under it.

COR. 1.-Since the product of any two factors will have as many figures or one less than both factors, so in division the number of figures of the divisor and quotient will either be equal to or one greater than that of the dividend.

COR. 2.-When the divisor is contained in the same number of figures of the dividend as is in the divisor, then the number of figures of the divisor and quotient will be one more than that of the dividend; but when it requires an additional figure of the dividend to contain the divisor, then the number of figures of the divisor and quotient will be equal to that of the dividend.

PROBLEMS.

1. Divide 9253360 by 2140; thus, (2.) 214|0) 925336|0 ( 4324

26432
856

104
693
4324

105728
642
2140

26432
513 17296

26432 ) 2748928 ( 104 428 4324

26432
856
8648

105728
856 9253360

105728

3. Divide 987654321 by 12300. 123|00 ) 987654321 ( 80297

80297 984

12300 365

240891 246

160594 1194

80297 1107

987653100 873

1221 861

987654321 1221, remainder. COR. 1.-When the dividend is not the exact product of two integral numbers, there will be a remainder, and the dividend is equal to the product of the quotient and divisor plus the remainder.

COR. 2.-When there are the same number of zeros in divi. dend and divisor, beginning with the order of units they may be canceled ; and when there are zeros in the divisor only, they may be omitted, and also the same number of figures in the dividend, which after the division is performed must be brought down as a part or the whole of the remainder.

EXAMPLES.

1. Divide 235643 by 123.
2. Divide 345678 by

234. 3. Divide 234567 by 891. 4. Divide 1357916 by 248. 5. Divide 369875432 by 1768. 6. Divide 487698425 by 625. ng Divide 987654321 by 1234. 8. Divide 876543219 by 2345. 9. Divide 678956732 by 1546. 10. Divide 34567890 by 2564. 11. Divide 786954321 by 176543. 12. Divide 678900432 by 1004000.

The student must not proceed until he is familiar with division.

EXAMPLES.

1. What cost 5 lbs. of sugar at 10 cts. per lb. ?

2. At 10 cts. per lb., how many lbs. can be bought for 50 cts. ?

3. What cost 10 lbs. of sugar at 10 cts. per lb. ?

4. At 10 cts. per lb., how many lbs. can be bought for 100 cts. ?

5. What cost 15 lbs. of sugar at 10 cts. per lb. ?

6. At 10 cts. per lb., how many lbs. can be bought for 150 cts.? Y. What cost 20 lbs. of

sugar at 10 cts. per lb. ? 8. At 10 cts. per lb., how many lbs. can be bought for 200 cts. ? 9. What cost 25 lbs. of sugar at 10 cts. per

Ib. ? 10. At 10 cts. per lb., how many lbs. can be bought for 250 cts. ?

11. What is the cost of 4 barrels of molasses at 18 dollars a barrel ?

a 12. What is the cost of 9 barrels of sugar at 15 dollars a barrel ?

13. What is the cost of 25 pounds of beef at 9 cents a pound ?

14. What is the cost of 33 pounds of butter at 21 cents a pound?

15. What is the cost of 147 pounds of cheese at 16 cents a pound ?

16. What is the cost of 123 barrels of cider at 4 dollars a barrel ?

17. What is the cost of 436 barrels of flour at 8 dollars a barrel ?

18. What is the cost of 432 tons of coal at 6 dollars a ton ?

19. In an orchard there are 9 rows of trees and 43 trees in each row; how many trees are there in the orchard ?

20. Bought 6 pieces of cloth, each containing 42 yards, at 6 dollars a yard; how many yards were there, and what did the whole cost?

21. Bought 9 pieces of cloth, each containing 43 yards, at 5 dollars a yard, and 25 barrels of flour at 6 dollars a barrel; what did the whole cost?

22. If 4 barrels of flour cost 32 dollars, what is that

per barrel ?

23. A man paid 320 dollars for two horses ; what was the price of each ?

24. A man owning 480 acres of land, wishes to divide it equally among his 4 sons: how many acres will

each get ?

25. A farm of 120 acres is laid off into six fields of equal size; how many acres in each?

26. A man is 45 years old, and he is three times as old as his son. How old is the son ?

27. A farmer sold 325 bushels of wheat for 650 dollars. What was the rate per bushel ?

28. In an orchard there are 625 trees, and there are 25 trees in a row. How many rows are there?

29. In a garden there are 100 heads of cabbage in ten rows, and an equal number in each row. How many heads in each row ?

30. A farmer sowed five bushels of clover-seed on forty acres of land. How many acres to a bushel of seed ?

31. A man dying, bequeathed his estate of 580,640 dollars, as follows: 60,000 dollars to each of two daughters, 125,000 dollars to a son, 22,000 dollars to each of his four sisters, and the remainder to his grandson. What was the grandson's share ?

32. Four persons enter into partnership; the first invests 8564 dollars, the second 500 dollars more than the first, the third as much as the first and second, and the fourth 645 dollars less than the third. What was the whole amount invested ?

33. Bought silk for 1600 dollars, lace for 960 dollars, shoes for 356 dollars, a shawl for 30 dollars, and some ribbon for 5 dollars. Paid at one time 500 dollars, at another 685 dollars, and at a third 800 dollars. How much is still owing ?

34. A man bought a carriage and two horses for 963 dollars; the horses were valued at 450 dollars. What