Note. In questions of this kind it is generally the simplest way to find what 1 article will cost, then it may easily be told how much any number will cost.

19. 4 men would do it in 1 half the time that 2 would do it. Or, you may say, if 2 men would do it in 6 days, 1 man would do it in 12 days, and 4 men in of that time, or 3 days.

SECTION VI.

A. 4. 2 halves of any number make the whole number. Therefore 2 is 1 half of 2 times 2; or 4. It is of 4 times 2, or 8.

Let the pupil answer these questions in the following manner: 4 is of 3 times 4; 3 times 4 are 12. 5 is of 7 times 5; 7 times 5 are 35.

B. 2. 4 is 2 times 2.

4. 6 is 2 times 3.

16. 2 thirds of any number is twice as much as of the same number. If 4 is of some number, then 1 half of 4 or 2 is of that number; is of 6; therefore 4 is of 6. 20. If 6 is of a number, of 6 or 2 is of the same number; 2 is of 8; therefore 6 is & of 8.

23. It is evident that of a pound will cost only of what will cost. If cost 6 cents, will cost 2 cents, and the whole pound 14 cents.

26. It will probably be perceived by this time, that of a number being given, it is necessary to find, and then the number is easily found; 4 being 4,2 is, and 2 is of 14. 45. 24 being, of 24 or 3 will be ; 3 is of 27.

C. 6. 20 being 4, 5 is 4, and 5 is times 6, and of 6.

D. 4. 18 is 3 times 6, and 6 is

Ans. 24 dollars.

of 35; and 35 is 5

of 4 times 6, or 24.

6. 54 is of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars.

7. 10 feet is of 15 feet.

8. If are under water, there must be
4 is of 12.

ter.
9. If are under water, there must be
6 is of 10.

ter.

out of the wa

out of the wa

10. and are . bear cherries and peaches; consequently, the 10 which bear plums must be the other ; 10 is 4 of 35. 10 bear peaches, and 15 bear cherries.

11., and, and, and, are ; therefore 12 must be the other of the whole. The whole number is 54.

Miscellaneous Examples.

6. The greyhound gains upon the fox 4 rods in a minate. It will take him 20 minutes to gain 80 rods.

8. of 24. Or you may say, 1 sheep would cost 3 dollars, and 3 sheep 9 dollars.

9. 30 horses will eat 10 times as much as 3 horses. 11. 10 dollars apiece, and 2 dollars a yard.

12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for 5 weeks.

14. It would take them 5 times as long to eat 40 bushels, as it would to eat 8 bushels.

15. 4 horses would eat 4 bushels in 3 days, and it would take them 9 times as long to eat 36 bushels. Ans. 27 days.

16. If 2 men spend 12 dollars in 1 week, 1 man will spend 6 dollars in 1 week, and 30 dollars in 5 weeks, and 3 men would spend 3 times as much, or 90 dollars.

17. The shadow of the staff is of the length of the staff, therefore the shadow of the pole must be the length of the pole. 18 feet is of 27 feet.

20. It would take 2 men 3 times as long to do it as it would 6 men.

23. 8 men would do a piece of work 1 half as large in 2 days, and it would take 2 men 4 times as long to do it, or 8 days.

28. He must sell it for 56 dollars in order to gain 16 dollars. 56 dollars is 7 dollars per barrel

29. It cost him 35 dollars, and he must sell it for 45 to gain 10 dollars; 45 dollars is 9 dollars a firkin. 30. Ans. 56 cents, see section VI.

33. If it would last 3 men 10 months, it would last 1 man 30 months, and 5 men 6 months.

34. There are 8 times 5 in 40; and since the other would build as many times 9 as the first does 5, he would build 8 times 9 or 72 rods.

SECTION VII.

A. 13. of 20 is 4, are 16; 16 being †, 2 is ; 2 is of 14, and 16 is & of 14.

16. of 28 are 12; 12 is 2 times 6, and 6 is 1 of 48 (12 is of 48), and 48 is 6 times 7 and 4 of 7.

B. 1. of 15 are 12; 12 is 6 times 2; 2 is is of 20); of 21 is 7; 20 is 2 times 7 and 2. of 18 are 24; 24 is of 27;

times 5 and 2 of 5.

of 35 is

of 20 (12 of 7.

5; 27 is 5

C. This article contains the multiplication table, in which the numbers from 10 to 20 are multiplied by the ten first numbers.

SECTION VIII.

A. 1. IN one there are 2 halves; in 2 there are twice as many halves, that is four halves or .

2. In one there are 2 halves, in 3 there are three times two halves, that is, 6 halves, or .

3. In one there are 3 thirds, in 2 there are twice 3 thirds, that is, 6 thirds, or §.

Each line

15. Draw two lines on the board thus call one; divide the upper line into two equal parts, each part is one half of one, or one half; divide the lower line in the same way, and then call for the answer. The form of the question may be varied by asking how many half apples there are in two apples.

37. In 5 and 2 thirds, how many thirds? Draw five parallel equal lines, divide them into thirds, and draw another parallel with the others and two thirds as long, let the scholar count the thirds. In all these cases of illustration on the board, it is presumed that the scholar will soon take the hint, and devise illustrations for himself at his seat.

SECTION IX.

A. 2. signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or 2.

6. 7 times is }, or 21.

B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 2, which added to 8 make 10.

18. 4 times 3 are 12, and 4 times are 12, or three whole ones, which added to 12 make 15.

32. 2 times 3 are 6, and 2 times

6 make 6.

are, which added to

40. 10 barrels at 3 dollars and a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times § is 50, or 8 and of a dollar. Ans. 382 dollars.

C. 2. to each would be 3 times, or, which are 21 oranges.

8. 1 or 2 bushels.

4. 7 times are 21, or 5 gallons.

5. 8 yards and or 2 yards, that is 10 yards. 6. 4 times 2 are 8, and 4 times

added to 8 make 10 bushels.

are 42, or 2, which

12. It would take 1 man 3 times as long as it would 3 Ans. 13 days.

men.

14. 3 men would build 3 times as much as 1 man; and in 4 days they would build 4 times as much as in 1 day. Ans. 388 rods.

15. Ans. 12 yards.

A. 21. of 1 is

of 4 ist, or 1. of 7 is 1, or 21. 27. of 1 is

SECTION X.

of 2 is 2 times as much, or }. of 5 is, or 13. of 6 is f, or 2.

of 2 is 3. of 3 is . of 7 is 7, or 13.

any

num

This manner of reasoning may be applied to ber. To find of 88: it is, for 4 of 38 is 38 times as much as of 1, and of 1 is, consequently of 88 is 3, and is 53.

40. To find

of a number,

must be found first, and then will be 2 times as much. of 7 is 1, and 2 times are, or 48.

74.

of 50 is 0, or 5; is 4 times as much; 4 times 5 are 20, 4 times & are 20, or 2, which added to 20 make 223.

Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Ans. 13 apiece.

3. of 3 is; of a bushel apiece.

4.

of 7 is 4; he gave away 41, and kept 21.

6. I half dollar a yard, or 50 cents.

7. of 7 is, or 13; which is 40 cents.

8. of 8 is 13.

of a dollar is

of 100 cents, Ans. 1 dollar and 40 cents a bushel. of 100 is 333. Ans. 1 dollar and 331

cents, or it is 1 dollar and 2 shillings.

9. If 3 bushels cost 8 dollars, 1 bushel will lars and, and 2 bushels will cost 5 dollars. lars and 2 shillings, or 332 cents.

13. If 7 pounds cost 40 cents, 1 will cost 5 pounds will cost 574 cents.

cost 2 dol

Ans. 5 dol

cents; 10

16. 1 cock would empty it in 6 hours, and 7 cocks would empty it in of 6 hours, or of 1 hour, which is of 60 minutes; of 60 minutes is 51 minutes.

SECTION XI.

conse

A. 2. 2 halves of a number make the number quently 1 and 1 half is the half of 2 times 1 an 1 half,

which is 3.

15. 4 is of 5 times 4 and 4, which is 224.

17. 4 is of 9 times 44, which is 394.

B. 4. 5 is 3 times of 5, which is ğ, or 13.

30. If 8 is

of some number, of 8 is of the same

number. of 8 is 2, 23 is of 4 times 23 which is 103; therefore 8 is

of 103.

of 8 of 94.

52. If of a ton cost 23 dollars, of a ton must be of