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 1.

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 & and of a dollar. Ans. 383 dollars.

C. 2. to each would be 3 times, or, which are 24 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 8 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. 388 rods.

15. Ans. 12 yards.

A. 21. of 1 is

of 4 is, or 11. of 7 is 1, or 2. 27. of 1 is

SECTION X.

of 2 is 2 times as much, or of 5 is f, or 1. of 6 is, or 2.

of 2 is . 1 of 3 is f. of 7 is 7, or 1f. This manner of reasoning may be applied to any number. To find of 38: it is 38, for of 38 is 38 times as much as of 1, and of 1 is, consequently of 88 is 4 and 38 is 54.

40. To find of a number, must be found first, and then will be 2 times as much. of 7 is, and 2 times Jare, or 48.

74.

of 50 is 5, or 5; is 4 times as much; 4 times 5 are 29, 4 times are 20, or 23, 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. 1 apiece.

3. of 3 is ; of a bushel apiece.

4.

of 7 is 4; he gave away 4, and kept 2. 6. I half dollar a yard, or 50 cents. 7. of 7 is 7, or

which is 40 cents. 8. of 8 is 13.

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

of 100 is 333. 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.

lars and 2 shillings, or 333 cents.

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

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.

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

15. 44 is of 5 times 4 and 4, which is 224.

17. 4 is of 9 times 44, which is 39.

B. 4. 5 is 3 times of 5, which is §, or 1.

30. If 8 is

of some number, of 8 is of the same number. of 8 is 23, 23 is of 4 times 23 which is 103; therefore 8 is of 103.

40. If 8 is, of 8 is; of 8 is, is of, or 94; therefore 8 is of 93.

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

23, that is 4g dollars, and the whole would cost 9 times as much, that is, 412.

69.of 55 is 7; 73 is of 5 times 7%, which is 36). 65 is of 361.

C. 4. 37 is of 32%, which taken from 37 leaves 43. Ans. 4 dollars.

5. 7 feet must be of the whole pole.

6. If he lost, he must have sold it for 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. Ans. 833 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and; of 60 minutes are 46,minutes and; of 60 seconds are 40 seconds. Ans. 10 hours, 46

minutes, 40 seconds.

10. Find of 33, and subtract it from 17. Ans. 31. 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.

14. Reduce 8 to an improper fraction; this is, in 8 and 3 three tenths, how many tenths?

83

Ans. 8.

B. 8. 22 are how many times 1? That is, in 28 sevenths how many whole ones? Ans. 34.

IX. B. 3. How much is 5 times 64? That is, how much is 5 times & and 4 sevenths? Ans. 329.

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. 62. B. 4. 12 is of what number? That is, 12 is 3 sevenths of what number? Ans. 28.

12. 4 is of what number? That is, 4 is 3 fifths of what number? Ans. 6.

SECTION XIII.

THE operations in this section are the reducing of fractions to a common denominator, and the addition and subtraction of fractions. The examples will generally show what is to be done, and how it is to be done.

4. It will readily be seen that and are 4.

25. In the fourth square of the second row, it will be seen that 1 half is ; and in the second square of the fourth row, is, both together make § and † make 7. 27. is the same as g

When these questions are performed in the mind, the pupil will explain them as follows. He will probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and 2 of 20 is 8, and of 20 is 2; therefore is, is, and is. All the examples should be explained in the same manner.

45. One whole one is §, one eighth of is. is 3 times as much, which is

51. 1 half is, and is, which added together make §. 61.is, is, is, which added together

make

67. is, is which added together make 1; from take, and there remains, or 1.

82. It will be easily perceived that these examples do not differ from those in the first part of the section, except in the language used. They must be reduced to a

common denominator, and then they may be added and subtracted as easily as whole numbers. is, and f is , and both together make 18 or 11.

86. is, and is . If be taken from, there remains

B. This article contains only a practical application of the preceding.

3. This example and some of the following contain mixed numbers, but they are quite as easy as the others. The whole numbers may be added separately, and the fractions reduced to a common denominator, and then added as in other cases, and afterwards joined to the whole numbers. 6 and 2 are 8; 1 half and are §, making in the whole 8 bushels.

5. 6 and 2 are 8; and and are 7 or 117, which joined with 8 make 917.

C. It is difficult to find examples which will aptly illustrate this operation. It can be done more conveniently by the instructer. Whenever a fraction occurs, which may be reduced to lower terms, if it be suggested to the pupil, he will readily perceive it and do it. This may be done in almost any part of the book, but more especially after studying the 13th section. Perhaps it would be as well to omit this article the first time the pupil goes through the book, and, after he has seen the use of the operation, let him study it.

SECTION XIV.

A. THIS section contains the division of fractions by whole numbers, and the multiplication of one fraction by another. Though these operations sometimes appear to be division, and sometimes multiplication, yet there is actually no difference in the operations.

The practical examples will generally show how the operations are to be performed, but it will be well to illus trate the operation for young pupils.

1 and 2. of is of the whole.

8 and 4.

ofis

16 and 17, of is of the whole