an entire unit, or whole one. 2d, explain the divisions into two, three, four, &c. parts. 3d, teach him to name the different parts. Make him observe that the name shows into how many parts one is divided, and how many parts are taken, in the same manner as it does when applied to larger numbers. for example, shows that one thing is to be divided into 7 equal parts, and 4 of those parts are to be taken. 4th, make the pupil compare the different parts together, and observe which is the largest. Ask him such questions as the following: Which are the smallest halves or thirds? Ans. Thirds. Why? Because, the more parts a thing is divided into, the smaller the parts must be.

A. 15. On plate II., count two squares in the sec ond row, and then ascertain the number of spaces or halves in them. There are 4 halves.

21. In the 2d row take 3 squares and 1 space in the 4th square; then count the spaces. Ans. 7 halves.

37. In the 3d row take 5 squares, and 2 spaces in the 6th; then count the spaces or thirds. Ans. 17 thirds.

54. In the 5th row take 6 squares, and 4 spaces in the 7th square; then count the spaces or fifths. Ans. 34 fifths.

B. 2. This operation is the reverse of the last In the 2d row count 4 spaces or halves, and see how many squares or whole ones it takes. It will take 2

It

38. In the 9th row count 48 spaces or 9ths, and see how many squares or whole ones it takes. will take 5 squares and 3 spaces in the 6th. 5 whole ones and .

Ans.

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 3.

6.7 timesis, or 2.

10. On the plate in the 3d row, 5 times which takes 3 squares and 1 space.

are,

Ans. 31.

24. In the 9th row take 4 spaces or 9ths, and repeat them 5 times, which will make 2, and wil take 2 squares and 2 spaces. Ans. 23.

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, or three whole ones, which added to 12 make 15.

32. 2 times 3 are 6, and 2 times are, which added to 6 make 69.

40. 10 barrels of cider at 3. dollars and a barzel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times is 50, or 8 and 2 of a dollar. Ans. 382 dollars.

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

10

3. or 2 bushels.

4. 7 times are 2, or 5 gallons.

5. 8 yards and 2 or 2 yards, that is, 10 yards. 6. 4 times 2 are S, and 4 times are 12, or 23, which added to 8 make 103 bushels.

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

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. 38 rods

15. Ans. 12 yards.

7 is 3, or 12.

This manner of reasoning may be applied to any number. To find 4 of 38: it is, for 4 of 38 is 38 times as much as of 1, and of 1 is, consequent y of 38 is 3, and 48 is 54. 40. To find of a number, and then will be 2 times as and 2 times are, or 43.

74. of 50 is 50, or 5; 4 times 5 are 20, 4 times added to 20 make 223.

must be found first,
much. of 7 is 1,

is 4 times as much
are 4o, or 2 wh

NOTE. The manner employed in example 40th is best for smah numbers, and that in the 74th for large numbers.

B. 2. Ans. 14 apiece.

3. of 3 is; of a bushel apiece.

4.3 of 7 is 41; he gave away 41 and kept 24. 6. 1 half dollar a yard, or 50 cents.

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

of a dollar is of 100 Ans. 1 dollar and 40 cents

8. of 8 is 1. of 100 is 334. Ans. 1 dollar and 333 cents, or it is 1 dollar and 2 shillings.

9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and, and 2 bushels will cost 5 dollars. Ans. 5 dollars and 2 shillings, or 333 cents.

13. If 7 pounds cost 40 cents, 1 will cost 54 cents; 10 pounds will cost 574 cents.

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 5' 7 minutes.

SECTION XI.

A. 2. 2 halves of a number make the number ; consequently 1 and 1 half is the half of 2 times 1 and 1 half, which is 3.

15. 4 is of 5 times 4 and, which is 224. 17. 4 is of 9 times 43, which is 399.

B. 4. 5 is 3 times of 5, which is 5, or 13. 30. If 8 is of some number, of 8 is

of the

same number. of 8 is 23, 23 is 1 of 4 times 23, which is 103; therefore 8 is 2 of 103.

40. If 8 is, of 8 is; of 8 is, is of or 9; therefore 8 is of 9 2.

52. If of a ton cost 23 dollars, of a ton mus be of 23, that is 43 dollars, and the whole would cost 9 times as much, that is, 413.

69.of 65 is 7; 73 is of 5 times 7%, which is 36. 65 is of 361.

៖

C. 4. 37 is of 323, which taken from 37 leaves 4 Ans. 4 dollars.

5. 7 feet must be of the whole pole.

6. If he lost, he must have sold it for 7 of what it cost. 47 is of 60%. Ans. 60 dollars and 424

cents.

Miscellaneous Examples.

1. The shadow of the staff is of the length of the staff; therefore the shadow of the pole is of the length of the pole. 67 is of 833. feet.

Ans. 83

It

2.9 gallons remain in the cistern in 1 hour. will be filled in 10 hours and 7; 7 of 60 minutes

are 46 minutes and ; & of 60 seconds are 40 seconds. Ans. 10 hours, 46 minutes, 40 seconds.

10. Find 3 of 33, and subtract it from 17. Ans. 34. 11. It will take 3 times 10 yards.

13. 5 is of 3; it will take as much. Or 7 yards, 5 quarters wide, are equal to 35 yards 1 quarter wide, which is equal to 11 yards that is 3 quarters wide.

15. of 37 dollars. 16. as much.

SECTION XII.

THE examples in this section are performed in precisely the same manner, as those in the sections to which they refer. All the difficulty consists in comprehending, that fractions expressed in figures signify the same thing as when expressed in words Make the pupil express them in words, and all the difficulty will vanish. Let particular attention be paid to the explanation of fractions given in the section.

VIII. A. 6. In 7 how many expressed in words, is, in 7 how many sixths? Ans. 42.

14. Reduce 8 to an improper fraction; that is, in 8 and 3 tenths, how many tenths? Ans. §.

B. 8. 23 are how many times 1? That is, in 23 sevenths how many whole ones? Ans. 34.

IX. B. 3. How much is 5 times 64? That is, how much is 5 times 6 and 4 sevenths? Ans. 324. V. & X. 15. What is of 27? That is, what is 5 eighths of 27? Ans. 167.

VI. & XI. A. 8. 7 is of what number? That is, 7 and 6 sevenths is 1 eighth of what number? Ans. 624.