thus till all the figures of the dividend have been brought down and annexed.

If, after dividing all the figures of the dividend, there is a remainder, it is to be written over the divisor and annexed to the quotient, as in short division.

PROOF. Multiply the quotient by the divisor, and to the product add the remainder; the sum should be equal to the dividend. Or,

Subtract the remainder, if there is any, from the dividend, and divide the difference by the quotient; the result should be equal to the original divisor.

NOTE 1. Commence the division by taking the fewest figures on the left of the dividend that will contain the divisor.

2. Place either a naught or some other figure in the quotient for every figure brought down from the dividend.

3. If in any case the product exceeds the partial dividend from which it is to be subtracted, the figure put in the quotient is too large, and must be diminished.

4. The remainder must always be less than the divisor.

5. Be careful to observe the rules (Art. 38) for pointing off decimals in the quotient.

EXAMPLES FOR PRACTICE.

1. Divide 306540106 by 13; by 14. (2.) By 15; by 16. (3.) By 17; by 18; by 19.

4. Divide 417870060 by 23; by 38; by 85. (5.) By 105; by 347.

6. Divide 1074083 by 37000. 37000) 1074.083 (29-1088

First remove the decimal point 3 places to the left; this divides it by 1000. Then divide the whole number, 1074, by 37; the figures pointed off are a part of the remainder. See note, Art. 39.

7. Divide 1074083 by 47500; by 7180. 8. By 94300; by 871000.

74

334

333

1083 rem.

37000

9. If 18 bags of coffee cost 486 dollars, what is 1 bag worth? 10. A man bought a field, containing 19 acres, for 11108 dollars; how much did he pay for each acre?

11. There are 320 rods in a mile; how many miles are there in 5104678 rods, and how many rods over?

12. If a rail-car can run 450 miles in 1 day, how far is that per hour?

13. What is 1 barrel of flour worth, if 350 barrels cost $2154.48?

In the 14th example there are 7 decimals used in the dividend, (counting all the decimal naughts which were annexed to the remainders,) and 3 in the divisor; or 4 more decimals in the dividend than in the divisor. The sign+to the right of the quotient shows that the division can be continued further.

In the 15th example, there are 11 decimal places in the dividend, and none in the divisor; consequently 11 decimals are required in the quotient. As there are not so many, as many naughts must be prefixed as are wanted to make the number. The sign to the right of the quotient figure indicates that the last quotient figure is a little too large.

16. How many times is 30.015 contained in 80.41607?

Carry the quotients in the 17th, 18th, 19th, and 20th examples to at least 7 decimal places, if there are remainders.

17. Divide 15 by .007; by 8.16. (18.) By .0104; by 7; by 900.

19. Divide .00005 by 270; by 6.85. (20.) By 4019; by 31.07; by 1000.

21. Divide $1000.00 equally among 187 men, getting the answer to the nearest mill.

22. Divide thirty dollars one cent equally among 13 men, as in the last example.

QUESTIONS. What is division? What terms are used in division? What is the dividend? The divisor? The quotient? The remainder? Give an example. What are factors of the dividend? What is the sign of division? Give examples showing how division may be expressed. If a number or quantity be divided into 2 equal parts, what is one of the parts called? If divided into 3 equal parts, what is one part called? 2 parts? Give examples, and show how such expressions as one half, one third, three fourths, &c., are written, Show by an example how the division of larger numbers is performed on the slate. What is to be done if the divisor is not contained in the first figure of the dividend? If after dividing any figure there is a remainder, what is to be done? Give an example. How is division to be performed when there are decimals in the dividend only? What may be done if, after dividing, there is a remainder, and more decimals are desired in the quotient? What is the general rule for decimals in division? What is to be done if the dividend has not so many decimal places as the divisor? How do you divide by 10, 100, 1000, &c.? Why? By 3000, 80000, &c.? Why? What is to be done with the figures pointed off at the right of the dividend, if decimals are not desired in the quotient? How are higher denominations reduced to lower? (See Art. 31, p. 34.) How are lower denominations reduced to higher? When should you divide by short division? When by long division? Perform an example by long division, and explain each step in the process. What is the method of proof? What remark is made in note 1? In note 2? Note 3 Note 4? Note 5? What What does the

does the sign + at the right of a quotient indicate?

at the right of a quotient indicate?

41. Review Art. 34, question 17 to 31, and Art. 36.

1. How many times is 4 contained in 135? Solution. 135 = 120 +15; 4 is contained in 120, 30 times; in 15, 34 times, which added to 30 makes 334.

2. How many times is 4 contained in 151? In 178? 369? 570? 3. How many times is 5 contained in 257? In 341 684 = 600 +50+34?

4. In 174 farthings how many pence? Solution. Since there are 4 farthings in 1 penny, there will be as many pence in 174 farthings as there are times 4 in 174. Ans. 43d. 2qr.

5. How many pence in 48 farthings? In 169 farthings? In 275 farthings?

6. How many shillings in 72 pence? In 124d.? 30s. 7d.

In 367d.? Ans.

In 47s.?

In 59s.?

In 84 oz.?

In 367 oz.?

In 56 gr.?

7. How many pounds in 25 shillings?
8. How many pounds in 36 oz. Troy?
9. How many dwt. in 24 gr.? In 32 gr.? In 48 gr.

10. How many oz. in 20 dwt. In 25 dwt.? In 80 dwt.? In 85 dwt.? In 109 dwt.?

11. How many oz. in 16 dr. Avoidupois? In 24 dr.? In 30 dr.? In 48 dr.? In 54 dr.?

12. How many lb. in 32 oz.? In 40 oz.? In 56 oz.? In 64 oz.? 13. How many quarters in 8 nails? In 19 na.? In 54 na.? In 160 na.? In 163 na.?

14. How many feet in 36 inches? In 48 in.? 72 in.? In 720 in.? In 725 in.?

In 54 in.? In

15. How many yards in 24 feet? In 240 feet? In 2400 feet? In 25 feet?

In 242 feet?

16. How many miles in 32 furlongs? In 320 fur. In 37 fur.? In 72 fur.? In 720 fur.?

17. How many square yards in 18 sq. feet? 1800 sq. feet? In 23 sq. feet? In 72 sq. feet?

18. How many cord feet in 16 cubic feet? cu. ft.? In 37 cu. ft.? In 45 cu. ft.?

In 180 sq. feet? In
In 76 sq. feet?
In 20 cu. ft.? In 32

In 57 cu. ft.?

19. How many cords in 8 cord feet? In 12 cord feet? In 24 C. In 48 C. ft.? In 480 C. ft.? In 72 C. ft.? In 720 C. ft.? In 784 C. ft.?

ft.

20. In 15 pints how many quarts? How many gallons?

21. How many quarts and gallons in 20 pints? In 24 pints? In 45 pints?

22. How many pecks in 48 quarts? How many bushels?

23. How many pecks and bushels in 64 quarts? In 75 quarts? In 96 quarts? In 120 quarts? In 125 quarts?

Questions to be performed by Division.

42. 1. How many bushels of wheat, at $0.875 per bushel, can be bought for $85.17?

2. How many tons of coal, at $7.75 per ton, can be bought for $186.00?

3. If 350 men are to share equally $10051, how much shall each have? Get the answer to the nearest cent.

4. How many cords of wood, at $6.50 per cord, can be bought for $135.85?

5. Bought 358 barrels of flour for $2159.25; what was the price per barrel?

6. In 1841983 farthings how many pence? In these pence how many shillings? In these shillings how many pounds? Do it by short division.

7. In 180451 mills how many cents? 8. In 3517047 mills how many cents?

How many dollars?
How many dollars?

Ans.

10. In 130433717 seconds how many min.? h.? da.? yr.?

60) 130433717 sec.

60) 2173895 min. 17 sec.

24) 36231 h. 35 min.

105 lb. 11 oz. 16 dwt. 23 gr. 365.25 ) 1509 d. 15 h.

Ans. 4 yr. 48 d. 15 h. [35 m. 17 sec.

11. In 4460175 minutes how many hours? days? years? 12. In 916000 grains apothecaries' weight how many pounds, ounces, &c.?

Reduce the grains to scruples, then these scruples to drams, &c. 13. In 8164096 drams Avoirdupois how many oz.? lb.? qr.? cwt.? tons?

14. In 2289600 inches how many ft.? yd.? rd,? fur.? m. 2 15. In 876510 feet how many miles, furlongs, &c.? Reduce the feet to yards, then these yards to rods, &c.?

16. In 681045 inches how many nails? quarters? yards? 17. In 31363200 square inches how many acres? 18. In 4817690 square yards how many square miles? 19. In 810650 cubic inches how many cubic yards? 20. In 6541 pints how many quarts? pecks? bushels? 21. In 68145 pints how many quarts? gallons?

22. In 4108000 seconds how many minutes? hours? days? 23. In 86417 cubic inches how many wine gallons? (59.) How many dry gallons, or half pecks? (58.)

Perform the following divisions, annexing decimal naughts to the dividend, if there are remainders, to 6 places of decimals, (24.) 18. (25.) }; }. (26.) ; f. (27.) §; 487.

(28.) ; 168; 18.

8

Perform the following divisions; if there is a remainder write it over the divisor and annex it to the quotient.

(29.) 167. Ans. 112. 1586. (30.) 10891; 20898.

(31.)

14

1883; 21768400. (32.) 5070901; 38187

80016

33. What number multiplied by 684 will make 7387884 ?