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40. To find f of a number, $ must be found first, and then j will be 3 times as much. of 7 is }, and 2 times į are y, or 44:

74. 3 of 50 is 50, or 55; $is 4 times as much ; 4 times 5 are 20, 4 times are 2, 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. lf apiece. 3. of 3 is ; of a bushel a piece. 4. of 7 is 4}; he gave away 41, and kept 24. 6. I half dollar a yard, or 50 cents. 7. } of 7 is }, or 13; of a dollar is of 100 cents, which is 40 cents. Ans. 1 dollar and 40 cents a bushel.

8. of 8 is 14 of 100 is 33. Ans. 1 dollar and 33% cents, or it is 1 dollar and 2 shillings.

9. If 3 bushels cost 8 dollars, I bushel will cost 2 dollars and y, and 2 bushels will cost 5f 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 57 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 514 minutes.

SECTION XI. A. 2. 2 halves of a number make the number, conse quently 1 and 1 half is the half of 2 times 1 and 1 half, which is 3.

15. 4$ is 1 of 5 times 4 and 4: which is 227.
17. 4} is of 9 times 4$, which is 399.
B. 4. 5 is 3 times of 5, which is , or 14.

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

40. If 8 is 6, 1 of 8 is 1; } of 8 is g, is of 68, or 94 ; therefore 8 is of 93.

52. If of a ton cost 23 dollars, $ of a ton must be t.of 23, that is 45 dollars, and the whole would cost 9 times as much, that iš, 414.

69. of 65 is 73 ; 7 is of 5 times 7%, which is 367. 65 is 3 of 364.

C. 4. 37 is f of 32%, which taken from 37 leaves 41 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 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 83%. Ans. 839 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and } ; 7 of 60 minutes are 468 minutes and $; $ of 60 seconds are 40 seconds. Ans. 10 hours, 46 minutes, 40 seconds.

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

13. 5 is. g 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 113 yards that is 3 quarters wide.

15. f 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 1 ? 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. i.

B. 8. ¥ 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. 329.

V. & X. 15. What is ģ of 27 ? That is, what is 5 eighths of 27? Ans.163.

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 f of what number? That is, 4 is 3 fifths of what number? Ans. 64.

SECTION XIII.

The operations in this section are the reducing of fractions to a common denominator, and the addition and Bubtraction 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 f 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, 1 is Ķ, both together makes and į make ž.

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 t of 20 is 2 ; therefore { is , is so, and it is zo. All the examples should be explained in the same manner. 45. One whole one is 56

58, one eighth of bed is a is 3 times as much, which is .

1 half is , and is, which added together make f. 61. is 2o, 3o is , is zo, which added together make 18

67. is 82,4 is eq, which added together make 13 ; from 1 take , and there remains 1, 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

51.

denominator, and then they may be added and subtracted as easily as whole numbers. f is 1%, and % is 6, and both together make 1 or 1: 86. is , and is s. If a 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 fare g, making in the whole 8. bushels.

5. 6 and 2 are 8; 1 and 1 and fare i7 or 117, which joined with 8 make 917.

C. It is difficult to find examples which wil} aptly illustrate this operation. It can be done more conveniently by the instructor. Whenever a fraction occurs, 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 omiť 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.

which may

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. of is 7 of the whole.
3 and 4. of is.
16 and 17. f of } is its 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, .

43. We cannot take ţ of 5 pieces, therefore we must take ţ of , which is Is, and is 5 times as much as , therefore { ofis :

78. 8} is 4, £ of 4 is } 79. 8 is , of 4 is g, consequently, 7 of ® is 4%, or 115 *86. We may say t of 89 is 2, and 24 over, then 24 is , and } of is 34, hence of 89 is 234. 90. 4 of 187 is 23}, and į is 3 times as much, or 734.

B. 4. It would take 1 man 4 times 94, or 37 days, and 7 men would do it in t of that time, that is in 52 days.

SECTION XV.

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

1. Since there are fin 2, it is evident that he could give them to 6 boys if he gave them $ apiece; but, if he gave them zapiece, 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. 68 is 27. If 1 of a bushel would last a week, 67 bushels would last 27 weeks ; but, since it takes 4, 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 a piece, he could give it to only } of that number, that is, to 5 persons, and he would have 1 bushel left, which would be f of enough for another.

23. 94 is , and 14, is 4. If it had been only 4 of a dollar a barrel, he might have bought 66 barrels for 98 dollars; but, since it was ų a barrel, he could buy only 11 of that number, that is, 6 barrels. 25 and 26. Ans. 94.

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