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23, that is 45 dollars, and the whole would cost 9 times as much, that is, 41%.

69. $ of 65 is 7'; 7is of 5 times 7%, which is 363. 65 is 3 of 36.

C. 4. 37 is ģ of 32%, which taken from 37 leaves 45. Ans. 41 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 7 of 604. 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. 833 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and b; 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. 33. 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. 3 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.

VII. A. 6. In 7 how many 2 ? expressed in words, is in 7 how many sixths ? Ans...

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

Ans.se

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B. 8. 48 are how many times 1 ? That is, in 23 seve enths how many whole ones? Ans. 34.

IX. B. 3. Ilow 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. 7f 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 i number? Ans. 63.

SECTION XIII. The operations in this section are the reducing of fracstions 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 1 and 1 are *.
25. In the fourth square of the second row,

it will be < seen that i half is ; and in the second square of the fourth row, $ is both together make g and make f.

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. 1 of 20 is 5, and of 20 is 8, and t of 20 is 2; therefore { is a l is 27, and is to. All the examples should be explained in the same manner.

45. One whole one is &, one eighth of is o f is 3 times as much, which is .

51. 1 half is g, and 1 is , which added together make a 61. & is oo, &is , is bo, which added together

67. is it is , which added together make 17; from take &, and there remains 19, 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

make

common denominator, and then they may be added and subtracted as easily as whole numbers. f is 1, and f is

, and both together make 1or 14 86. } is , and 1 is . If} be taken from X, there re

mains :

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 į areš, making in the whole 85 bushels.

5. 6 and 2 are 8; and and fare for 127, which joined with 8 make 947

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 gocs 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.
3 and 4. of Jis
16 and 17. f of } is its of the whole.

83. 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

43. We cannot take of 5 pieces, therefore we must take of }, which is to, and f is 5 times as much as , therefore of isto

78. 8is 4*, & of 4 is 7. 79. 87 is een of 4 is to, consequently of bel is ff, or 115.

86. We may say 1 of 84 is 2, and 24 over, then 24 is 44, and 1 of 4 is 3%, hence of 84 is 234.

90: 4 of 18% is 27, and is 3 times as much, or 7{.

B. 4. It would take 1 man 4 times 98, or 37$ days, an 1 7 men would do it in of that time, that is in 54 days.

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SECTION XV. A. TAIS section contains the divisions of whole numbers by fractions, and fractions by fractions.

1. Since there are f in 2, it is evident that he could give them to 6 boys if he gave them fapiece; but, if he

gave them s 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. 61 is . If I 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 f 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 i bushel left, which would be g of enough for another.

23. 94 is , and 14 is 4. If it had been only of a dollar a barrel, he might have bought 66 barrels for 99 dollars; but, since it was 4 a barrel, he could buy only is of that number, that is, 6 barrels.

25 and 26. Ans. 9.

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81. 41 is 41, and 9f is 4. Now } is contained in 4 48 times, and 4 is contained only ti part as many times, consequently only 24 or 24.

B. i. is fi consequently, 5 pounds can be bought for of a dollar.

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

5. 1 is it, and j is it. If a pound had costs of a dollar, 14 pounds could be bought for of a dollar; but since it costs of, only } as many can be bought; that is, 43 pounds.

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

12. is , and f is Wit is contained in 14 15 times, but it is contained only 4 as many times; that is, 34 times.

Miscellaneous Examples. 5. of a penny is of 4 farthings. Ans. 24 farthings

. 6. of 12 pence. Ans. 10 pence. 7. } of 4 quarters is 2 quarters and 7 of a quarter ; ; of a quarter is f of 4 nails, which is 1f nails. Ans. 2 quar ters, if nails.

19. of 24 hours is 15 hours.

14. of 24 hours is 14 hours and of an hour; 4 of 60 minutes is 24 minutes. Ans. 14 hours, 24 minutes.

28. There being 4 farthings in a penny, 1 farthing is 7 part of a penny.

30. 3 farthings iş te of a penny.

31. 1 penny is to of a shilling, because there are 12 % pence in a shilling

84. 5 pence is of a shilling:
41. 1 shilling is the of a pound.
43. 3 shillings is at of a pound.
48. 1 farthing is I of une shilling.

49. 2 farthings is af, or it of a shilling. 5 farthings is dy of a shilling

51. 1 penny is to of 1 pound. 7 pence is zo of 1.6. 59. 3s. 5d. is 41 pence, which is a's of 1£.

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