9. Divide 3628497 by 37, by the abridged method, and prove.

10. Divide 1284634 by 96 by the same.

11. Divide 47389256 by 375; also by 284 and by 763. 12. Divide 83245796 by 2458; also by 372 and by 815. 13. Divide 6529374 by 3275; also by 4762.

14. Divide 5138267 by 789; by 426; and by 738.

15. Perform the same operations with the figures of each dividend reversed; and again, with the figures of each divisor reversed, the dividend carried to three decimal places when necessary.

Remark. The division can be continued as far as may be thought proper, if there be a remainder after the integers in the dividend are exhausted, by adding ciphers to it, provided a separatrix be placed in the quotient after the units resulting from the division of the integers. Why?

The method of Long Division is commonly used where the divisor exceeds 12; but, if the class has been properly trained in Oral Arithmetic, it will be found advantageous to use Short Division in all cases where the divisor does not exceed two, or even three figures. In practice, it will be found sufficiently easy, and will essentially aid the power of abstraction.

16. Divide 23569248 by every number containing two significant figures between 12 and 100, by short division, and prove. 17. Divide 62543965, and also 34902054 severally by the same, and prove.

18. Divide each of the above two numbers severally by 126, 135, 224, 364, and 452, and prove.

Exemplifications for the Black-board,

Where ciphers occur in the Divisor and Dividend, or in the Divisor alone.

19. Divide 568400 by 2600; and 673428 by 2400.

No. 1. Dividend, 5684100(26100 Divisor.

Remainder,

48 2181688 Quotient.

224

1600 568400 Proof.

No. 2. Dividend, 6734 28(24/00 Divisor.

193 2801438 Quotient.

Remainder, 1428

673428 Proof.

Suggestive Questions. — If a number should be divided into 4 equal parts, and each of these into 3 equal parts, into how many equal parts would the number be divided? If a number should be divided into 4 equal parts, and each of these again into 100 equal parts, into how many equal parts would the number be divided? 4 and 100 are factors of what number? Is the same result obtained, then, when a number is divided by factors taken separately, as when divided by their product? Into how many equal parts is dividend No. 1 divided by cutting off two figures at the right? See Principle 2, p. 123. What, then, is the quotient of 568400 by 100? What factor remains to divide 5684 ? What is the quotient when it is divided by 26? To what ranks of the original dividend does the undivided remainder 16 belong? Is it 16, then, or 1600 ?- -What is the quotient of dividend No. 2 by 100 ? What is the remainder? What other factor should divide 6734? What is the quotient of that division? What is the remainder? To what ranks of the original dividend does this undivided remainder belong? Is it 14, then, or 1400? Is the 28 also an undivided remainder? To what ranks do the 2 and the 8 belong? What, then, is the whole undivided remainder of the original dividend?

20. Divide 14260 by 530; 726500 by 670; 257600 by 3400; and 8276000 by 270, and prove.

21. Divide 265023 by 610; 806284 by 7300; and 52648 by 70, and prove.

Exemplifications for the Black-board, Where Decimal Fractions occur in the Dividend or Divisor, or

in both.

22. Divide 24 by 6; 24 by '6; 24 by '6; 24 by '6; and

24 by 6.

No. 1.

6)2.4

$4

Suggestive Questions. - How many decimal places in the dividend, or product, of No. 1? How many in the divisor factor? As the number of decimal places in the factors, then, must correspond with the number in the product (See Multiplication, pp. 153, 154), how many decimal places are required in the quotient factor? How many are required in No. 2,

then? Why? How many in No. 3? Why? How many decimal places are there in the divisor factor of No. 4? How many ought there to be in the dividend product, then? How shall that be supplied without altering its value? See p. 118, l. 27. How many decimal places are there in the dividend product of No. 5? Are there any in the divisor factor? How many should there be in the quotient factor, then? How shall the second one be supplied? Is there more than one decimal place in this number, '40? Where should the cipher be placed, then, in quotient factor of No. 4 ? 23. Divide 325 76 by 23'4; 589 42 by '72; 68945 by 7.32; 89728 by '8; and 56 238 by 62, and prove by multiplication.

Practical Exercises.

1. If a piece of work can be accomplished by 1 man in 36 days, how long should 4 men be about it? Prove.

2. If 5 men can do a piece of work in 4 days, how many days should 1 man take to do it? How many days should 4 men? Prove.

3. If 12 barrels of flour cost $72, what will 1 barrel cost? What will 5 cost? Prove.

4. If 5 pounds of brown sugar cost 35 cents, what will 12 pounds cost? [In this and several exercises that follow, the leading question, what will 1 cost, weigh, &c., is omitted, but can readily be supplied by the teacher, if it is found necessary, or, still better, by the pupil himself.]

5. If $420 be paid for 60 acres of land, what will be the cost of 45 acres at the same rate? Prove.

6. If 32 yards of cotton cloth cost $256, what will be the cost of 5 yards at the same rate?

7. If $60 gain $3'60 interest in one year, what will $100 gain in the same time? Prove.

8. If 6 yards of broadcloth cost $18, what will 15 yards cost? Prove.

9. A man dying, left a widow and 6 children, without a will. In such cases the law directs that the widow shall receive one third of the property for life, and that the remainder shall be equally divided among the children. The estate was valued as follows: a farm, at $6000; a yoke of oxen, $90; 3 horses, at $75 dollars each; 16 cows, at $24 each, 8 young cattle, at $10 each; 100 sheep, at $2.50 each; farming tools, $100; household furniture, $300; grain and provisions, $62. What was the share of the widow, and of each child?

EVOLUTION.*

Or Division into Two or more Equal Factors.

EXTRACTION OF THE SQUARE ROOT, OR DIVISION INTO TWO EQUAL FACTORS.

a. Pointing off squares into periods of two figures.

Exemplifications for the Black-board.

1. Write in columns, on the slate or black-board, as follows, the squares of '01, 1, 1, 10, 100, 1000, being the smallest significant figure, and the squares of '09, 9, 9, 90, 900, 9000, the greatest significant figure. Write, also, a line of ciphers, and point them off into periods of two figures, as under:

*Evolution is generally placed near the close of the book in treatises on arithmetic. But, as it is strictly an elementary process, and a mere branch of division, it comes much more appropriately here. And, if the preceding part of the book has been thoroughly mastered, the pupil will find no difficulty in extracting either the square or the cube root. Teachers who dislike the arrangement, however, can easily omit evolution until the review of the whole book.

of

Suggestive Questions. Of how many figures does the square any number of units consist? See squares of 1 and 9 above. Which period, then, will the square of units оссиру ? How many ciphers are there in the square of any number of tens? See the square of 10 and of 90 above. Which period, then, will be occupied by the significant figures of the square of tens? Which period will be occupied by those of the hundreds ? Of the thousands? &c. Which period will be occupied by those of the tenths? Ans. The period to the right of that of the

Which period by those of the hundredths? In which period, then, should you look for the root of the units? The root of the thousands? Of the tenths? Of the hundreds ? Of the hundredths? Why, then, do we divide numbers whose roots are sought into periods of two figures?

b. To find the Square Root, when it consists of tens ard

units.

2. What is the square root of 2916? Prove.

Suggestive Questions. What is the greatest square in 29? What is its root? Is this root 5 units or 5 tens? Deducting, then, the square of 50 from the given square, as above, what must be found in the remainder? See the exemplification in Involution, No. 1, SQUARE of 24, p. 165, or see 10th principle, p. 166. Which of these numbers is now known? Ans. Twice the When a product [416] and one of its factors [twice 50] is known, how can the other factor be found? See p. 57, 5. What should the remainder [416] contain, besides twice the product of the tens and units of the root? Is 2916 an exact square, then?