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 43.

74.

of 50 is 50, or 55; is 4 times as much; 4 times 5 are 20, 4 times are 20 or 23, which added to 20 make

223.

99

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 43, and kept 24.

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

7. 1 of 7 is 7, or 1; 3 of a dollar is 3 of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 8. 1 of 8 is 12. 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, I bushel will cost 2 dollars and, and 2 bushels will cost 5 dollars. Ans. 5 dollars and 2 shillings, or 33 cents.

13. If 7 pounds cost 40 cents, 1 will cost 5 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 51 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 44, which is 399.

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

of some number, of 8 is of the same number. of 8 is 24, 24 is of 4 times 23 which is 103; therefore 8 is of 103..

40. If 8 is, af 8 is 4; 1 of 8 is, is of 58, or 93; therefore 8 is of 92.

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

23, that is 48 dollars, and the whole would cost 9 times as much, that is, 413.

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

C. 4. 37 is of 328, which taken from 37 leaves 41 Ans. 4 dollars.

5. 7 feet must be 2 of the whole pole.

6. If he lost, he must have sold it for 7 of what it cost. 47 is 7 of 60%. Ans. 60 dollars and 42 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 834. Ans. 83 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7; of 60 minutes are 466 minutes and of 60 seconds are 40 seconds. Ans. 10 hours, 46

minutes, 40 seconds.

10. Find 2 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 112 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. 4 42.

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

B. 8. 2 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 2.

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 of 20 is 8, and of 20 is 2; therefore is, is 20, and is 2. All the examples should be explained in the same manner.

56

10

45. One whole one is 8, one eighth of is. is 3 times as much, which is .

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

61.

2 is

8

20, 1 is 201

is5

which added together

make ਨੂੰ ਨੰ 67. is, is, which added together make }; 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

is, and

denominator, and then they may be added and subtracted

as easily as whole numbers. both together make 1 or 13. 86. is 2, and is . If mains.

is 1, and

be taken from, there re

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 37 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 instructor. 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 illustrate the operation for young pupils.

1 and 2. ofis 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. signify 9 pieces or parts, and it is evident that

39.

of 9 parts is 3 parts, that is, .

43. We cannot take

of 5 pieces, therefore we must take of, which is, and is 5 times as much as †, therefore of is.

B. 4. It would take 1 man 4 times 9, or 375 days, and 7 men would do it in of that time, that is in 518 days.

SECTION XV.

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, or 10 months.

7. 63 is 27. If of a bushel would last a week, 6o 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.

23. 9 is 5, and 14 is . If it had been only of a dollar a barrel, he might have bought 66 barrels for 9 dollars; but, since it was a barrel, he could buy only of that number, that is, 6 barrels. 25 and 26. Ans. 94.