When the steps in the solution of an example are written, the process is called Long Division.

79. From the solution of the example given, the following are seen to be the successive steps taken in solving an example in Long Division:

Write the divisor at the left of the dividend, with a curved line between them.

Find how many times the divisor is contained in the fewest figures on the left hand of the dividend that will contain it, and write the quotient at the right of the dividend or above it. Multiply the divisor by this quotient, and place the product under the figures divided. Subtract the result from the partial dividend used, and to the remainder annex the next figure of the dividend.

Divide as before until all the figures of the dividend have been annexed to the remainder.

If any partial dividend will not contain the divisor, write a cipher in the quotient, then annex the next figure of the dividend, and proceed as before.

If there is a remainder after the last division, write it after the quotient, or with the divisor under it as part of the quotient.

PROOF. Multiply the divisor by the quotient, and to the product add the remainder, if any. If the work is correct, the result will equal the dividend.

1. The quotient figure may generally be readily obtained by finding how many times the first figure of the divisor is contained in the first *figures of the dividend that will contain it, making allowance for the addition to the product from the second figure of the divisor.

2. If the product of the divisor by the quotient figure is greater than the partial dividend from which it is to be subtracted, the quotient figure is too large.

3. Each remainder must be less than the divisor; otherwise the quotient figure is too small.

4. When there is no remainder, the divisor is said to be exact.

Divide each of the following numbers by 21, 32, 43,

Divide each of the following numbers by 37, 28, 19,

REVIEW EXERCISES.

ORAL EXERCISES.

81. 1. A bookseller bought 15 books for $30 and sold them at a gain of $ 15. How much apiece did he get for them? 2. A miller sold 8 barrels of flour at $5 a barrel and bought with the proceeds cloth at $ 4 per yard. How many yards did he purchase?

3. The cost of 10 tons of coal was $ 50, but I sold it at a gain of $20. How much did I receive per ton?

4. A bookseller bought 12 dictionaries at $8 apiece and sold them at $10 apiece. How much did he gain?

5. A steam yacht ran up a river for 8 hours at 10 miles per hour, and it came down in 5 hours. What was its rate per hour coming down?

6. The cost of 5 tons of coal at $5 per ton was $7 less than the cost of 8 cords of wood. What did the wood cost per cord?

7. The expense of sending delegates to a convention. was $65. If the entire expense had been $1 less, the expense to each member would have been $2. members were there?

How many

8. A man owed a debt amounting to $84. If he paid $60 of the sum in 5 months, in how long a time will he pay it all at that rate?

9. A farmer sold a grocer 12 pounds of butter at 20¢ a pound and took in exchange brooms at 24 cents apiece. How many brooms did he get ?

10. Two boys set out from the same place and traveled in the same direction, Arthur traveling 5 miles per hour and Bennie 3 miles per hour. How soon will Arthur be 18 miles ahead of Bennie?

11. How far has each traveled when Arthur is 24 miles ahead?

12. Suppose they travel in opposite directions at those rates, how far apart will they be in 8 hours?

13. In how many days can 4 men do as much work as 8 men can do in 6 days?

14. Howard's father gave him an acre of land to till so that he might earn his own pocket money. He sold the products of the land for $38, but he paid $5 for seeds to plant and $2 for help. How much money had he left?

15. A boy bought 6 lead-pencils for 24 and exchanged them for some erasers at 3¢ each. How many erasers did he get?

16. A man bought a carriage for $ 120. After spending $40 in repairing and painting it, he sold it for $150. Did he gain or lose, and how much?

17. A newsboy bought 20 papers for 30 cents and sold them at a gain of 30 cents. For how much apiece did he sell them?

18. Henry bought 6 car-tickets for 25 cents. How many could he buy for 50 cents?

19. A clerk saved $20 a month. In how many months could he save enough to buy a library worth $ 240 ?

20. A jeweler bought 12 watches for $ 72 and sold them at a gain of $3 each. At what price did he sell each of them?

21. The wages of a machinist were $21 per week, and his expenses were $10. In how many weeks can he pay a debt of $75 and deposit $ 46 in a savings bank?

WRITTEN EXERCISES.

82. 1. If a man paid $7564 for 124 horses, how much did each cost?

2. At $9 a yard, how many yards of velvet can I buy for $207?

3. How many weeks are there in 1736 days?