34. If a vessel sails 169 miles in one day, how far will she sail in 144 days? Ans. 24336 miles. 35. What will 698 barrels of flour cost at 7 dollars a barrel?

Ans. $4886.

36. What will 876 lbs. of sugar cost at 13 cents a pound?

Ans. 4888 cts.

37. What will 97 lbs. of tea cost at 93 cents a pound?

Ans. 9021 cts.

38. If a regiment of soldiers consists of 1128 men, how many men are there in an army of 53 regiments? Ans. 59784.

39. What will an ox weighing 569 pounds amount to at 8 cents a pound? Ans. 4552 cts. 40. If a barrel of cider can be bought for 93 cents, what will 75 barrels cost? Ans. 6975 cts. 41. If in a certain factory 786 yards of cloth are made in one day, how many will be made in 313 days? Ans. 246018 yds. 42. A certain house contains 87 windows, and each window has 32 squares of glass, how many squares are there in the whole house? Ans. 2784 squares. 43. There are 407 wagons each loaded with 30009 pounds of coal, how many pounds are there in the whole ?

Ans. 12213663 pounds. 44. Multiply three hundred and seventy-five millions two hundred and ninety-six thousand three hundred and twentyone, by seventy-nine thousand and twenty-four.

Ans. 29657416470704.

45. What would be the cost of 687 fother of lead at 78 dollars a fother? Ans. $50151.

SECTION V.

MENTAL OPERATIONS IN DIVISION.

In 6?

In 8?

In 10?

In 9?

In 12?

In 15?

In 14?

In 16?

In 18?'

In 21?

In 24?

In 27 ?

1. How many times 2 are there in 4? How many times 3 are there in 6? How many times 2 are there in 12? How many times 3 are there in 18? How many times 2 are there in 10? How many times 4 are there in 12? In 16? How many times is 2 contained in 3? In 7? How many times is 5 contained in 10? how many times? 4 in 12 how many times? 3 in 12 how many times? 6 in 12 how many times?

2 is

2. James had 8 apples, and John had half as many, how many had he?

3. How many pears at 2 cents a piece can you buy for 4 cents?

4. If you have 8 apples to give to 4 boys, how many can you give to each?

5. Joseph has 12 pears, which he wishes to distribute equally between 6 of his companions, how many can he give them apiece?

6. If a man travel 3 miles in an hour, how many hours will it take him to travel 12 miles?

7. Fourteen are how many times 2?

8. At 6 cents apiece, how many oranges can you buy for 18 cents?

9. If 1 pencil cost 2 cents, how many can you buy for 20 cents?

10. If you can buy 1 slate for 6 cents, how many can you buy for 30 cents?

11. If you had 60 cents, how many quires of paper could you buy at 20 cents per quire?

12. If one pound of raisins cost 12 cents, how many pounds may be bought for 60 cents?

13. If 10 dollars will buy 1 ton of hay, how many tons may be bought for 30 dollars?

14. 24 are how many times 4? are how many times 6? are how many times 8? are how many times 12?

15. 27 are how many times 3? how many times 9? 16. 28 are how many times 4? how many times 7?

17. 30 are how many times S? how many times 5? how many times 15? how many times 6? how many times 10?

13. 32 are how many times 4? 8? 16? 5? 9? 11? 13? 17? 19. 36 are how many times 4? 6? 9? 12? 5? 7? 11? 30? 35? 20. 40 are how many times 4? 5? 8? 10? 3? 6? 7? 9? 11? 21. 42 are how many times 6? 7? 14? 21? 8? 9? 10? 11? 22. 48 are how many times 4? 6? 8? 12? 5? 6? 7? 9? 10? 23. 56 are how many times 4? 7? 8? 9? 10? 11? 12? 18? 24. 60 are how many times 3? 4? 5? 6? 10? 12? 7? 8? 9 ? 25. 64 are how many times 4? 8? 16? 5? 6? 7? 9? 10? 11? 26. 66 are how many times 2? 3? 6? 22? 38? 4? 5? 7? 8? 27. 72 are how many times 8? 9? 12? 6? 7? 8? 17? 18? 28. 84 are how many times 7? 12? 8? 9? 10? 11? 13? 14? 29. A man buys 6 sheep for 48 dollars; what did he give apiece?

30. James gave 72 cents for 8 quarts of cherries; what were they a quart?

31. Thomas gave 96 cents for 8 penknives; what was the price of each?

32. Benjamin had 37 nuts; he gave 7 to James, and divided the remainder equally between himself and two sisters; how many will be his share?

33. Thomas has 98 cents; he gave his sister Nancy 18 and divided the remainder between himself and 9 others; how many will each have?

34. Daniel has 47 apples; he gave William 7 and Samuel 10; how many will he have, if he share the remainder with himself and 2 brothers?

Let the pupil perform the last 16 questions on the slate, and hence notice, that

DIVISION is a short or compendious way of performing Sub-* traction.

Its object is to find how many times one number is contained in another. It consists of four parts; the dividend or number to be divided; the divisor or the number to be divided by; the quotient, which shows how many times the divisor is contained in the dividend; and the remainder, which is always less than the divisor, and of the same name of the dividend.

When the divisor is less than 13, the question should be performed by

SHORT DIVISION.

EXAMPLE.

1. Divide 948 dollars equally between 4 men.

Dividend.

Divisor 4)948
Quotient 237

In performing this question, inquire how many times 4, the divisor, is contained in 9, which is 2 times, and 1 remaining; write the 2 under the 9 and suppose 1, the remainder, to be placed before the next figure of the dividend, 4, and the number would be 14. Then inquire how many times 4, the divisor, is contained in 14. It is found to be 3 times and 2 remaining. Write the 3 under the 4, and suppose the remainder, 2, to be placed before the next figure of the dividend, 8, and the number would be 28. Inquire again how many times 28 will contain the divisor. It is found to be 7 times, which we place under the 8. Thus we find each man receives 237 dollars. From the above illustration, we deduce the following

RULE.

See how many times the divisor may be had in the first figure or figures of the dividend, and place the result immediately under that figure, and what remains suppose to be placed directly before the next figure of the dividend, and then inquire how many times these two figures will contain the divisor, and place the result as before; and so proceed until the question is finished.

When the divisor exceeds 12, the operation is performed by

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

OPERATION.

Dividend.

The object of this question is to find, how many

Divisor. 17) 3978 ( 234 Quotient. times 3978 will contain 17,

34

17

57

1638

51

234

68 3978 Proof.
68

00 Remainder.

or how many times must 17 be subtracted from 3978, until nothing shall remain. We first inquire, how many times the first two figures of the dividend will contain the divisor; that is, how many times 39 (thousand) will contain 17. Having found it to be 2 times, we write 2 in the quotient and multiply it by the divisor, 17, 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 before the dividend, and inquire how many times it is contained in a competent number of figures in the dividend, and place the result in the quotient; multiply the figure in the quotient by the divisor, and place the product under those figures in the dividend, in which it was inquired how many times the divisor was contained; subtract this product from the dividend, and, to the remainder, bring down the next figure of the dividend, and then inquire how many times this number will contain the divisor, and place the result in the quotient, and proceed as before, until all the figures in the dividend are brought down.

NOTE 1.

It will sometimes happen, that, after a figure is brought down, the number will not contain the divisor; a cipher is then to be placed in the quotient, and another figure is to be brought down, and so continue until it will contain the divisor, placing a cipher each time in the quotient.

NOTE 2. The remainder in all cases is less than the divisor, and of the same denomination of the dividend; and, if at any time, we subtract the product of the figure in the quotient and the divisor from the dividend, and the remainder is more than the divisor, the figure in the quotient is not large enough.

PROOF.

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

To prove it by Multiplication, the divisor must be multiplied by the quotient, and to the product, the remainder must be added, and if the result be like the dividend, the work is right. See Example 1.

To prove it by Addition— Add up the several products of the divisor and quotient with the remainder, and if the result be like the dividend the work is right. See Example 2.

To prove it by casting out the 9's-Cast the 9's out of the divisor, and place the remainder at the left hand of a cross; then cast them out of the quotient, and place the remainder at the right hand of the cross, and lastly subtract the remainder from the dividend, and cast the 9's out of what may remain, and place the result at the top of the cross; and if it be like the product of the figures at the sides of the cross (after the 9's are cast out of their products) the work is right. See Example 3.

To prove it by Division itself, subtract the remainder from the dividend, and divide this number by the quotient, and the quotient found by this division, will be equal to the former divisor, when the work is right. See Example 4.