23, that is 4g dollars, and the whole would cost 9 times as much, that is, 413.

69.of 65 is 73; 73 is 1 of 5 times 73, which is 361. 65 is of 361.

C. 4. 37 is of 32%, which taken from 37 leaves 44. Ans. 4 dollars.

5. 7 feet must be 2 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?

Ans. 8.

10.

B. 8. 28 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. 32.

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. 63.

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 .

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 I make f 27. is the same as §.

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 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 3.

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

61.

make

67.

is, is, which added together make 11; from take, and there remains 13, 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 is , and both together make or 1.

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 8ğ 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 illus trate 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.

3 and 4.

16 and 17.

of is of the whole.
of is

of is of the whole.

33. Since of a share signify 3 parts of a share, it is evident that of the three parts is 1 part, that is 1.

39. signify 9 pieces or parts, and it is evident that of 9 parts is 3 parts, that is 4.

43. We cannot take of 5 pieces, therefore we must takeof, which is, and is 5 times as much as †, therefore of is

78. 83 is 42, of 42 is 7.

79. 84 is, of is, consequently of is 8, or 1H.

86. We may say of 8 is 2, and 24 over, then 24 is 2a, and of 22 is, hence of 83 is 234.

90. of 188 is 23, and 4 is 3 times as much, or 7}}.

B. 4. It would take 1 man 4 times 9, or 375 days, an 1 7 men would do it in of that time, that is in 5

SECTION XV.

days.

A. THIS section contains the divisions of whole numbers by fractions, and fractions by fractions.

1. Since there are in 2, it is evident that he could give them to 6 boys if he gave them apiece; but, if he gave them apiece, he could give them to only one half as

many, or 3 boys.

5. If of a barrel would last them one month, it is evident that 4 barrels would last 20 months; but, since it takes of a barrel, it will last them but one half as long, dr 10 months.

7. 6 is 27. If of a bushel would last a week, 68 bushels would last 27 weeks; but, since it takes, it will last only of the time, or 9 weeks.

13. If he had given of a bushel apiece, he might have given it to 17 persons; but, since he gave 3 halves apiece, he could give it to only of that number, that is to 5 persons, and he would have 1 bushel left, which would be of enough for another.

. If it had been only of a have bought 66 barrels for 98 a barrel, he could buy only

23. 94 is, and 14 is dollar a barrel, he might dollars; but, since it was of that number, that is, 6 barrels. 25 and 26. Ans. 94.

81. 4 is 21, and 9 is 48. Now 48 times, and 2 is contained only consequently only 2 or 24.

is contained in part as many times,

B. 1. is; consequently 5 pounds can be bought for of a dollar.

3. is, and is. If he had given only apiece, he could have given it to 9 persons; but, since he gave he could give it to only 1 half as many, or 4 persons.

5. is, and is 14. If a pound had cost of a dollar, 14 pounds could be bought for of a dollar; but since it costs, only as many can be bought; that is, 4 pounds.

9. is 8, and 1 is 5. If a bushel had cost of a dollar, 65 bushels might have been bought; but, since it cost 18, only part as much could be bought; that is, 4 bushels.

but

12. is, and is ; is contained in 1 15 times, is contained only as many times; that is, 3 times. Miscellaneous Examples.

5.

6.

7.

of a penny is of 4 farthings. Ans. 23 farthings. of 12 pence. Ans. 10 pence.

of 4 quarters is 2 quarters and of a quarter; } of a quarter is of 4 nails, which is 1g nails. Ans. 2 quar ters, 1 nails.

13. of 24 hours is 15 hours.

14. of 24 hours is 14 hours and of an hour; of 60 minutes is 24 minutes. Ans. 14 hours, 24 minutes. 28. There being 4 farthings in a penny, 1 farthing is part of a penny. 30. 3 farthings is 31. 1 penny is pence in a shilling. 34. 5 pence is 41. 1 shilling is

of a penny.

of a shilling, because there are 12

of a shilling.

of a pound.

43. 3 shillings is

of a pound.

48. 1 farthing is

of one shilling.

49. 2 farthings is, or

of a shilling. 5 farthings is

of a shilling.

240

59. 3s. 5d. is 41

51. 1 penny is of 1 pound. 7 pence is 1 of 1£.

41 240

pence, which is 4 of 1£.