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SECTION IX.

A. 2. į signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or .

6. 7 times į is 7, or 21.

10. On the plate in the 3d row,5 times are yo which takes 3 squares and 1 space. Ans. 33.

24. In the 9th row take 4 spaces or 9ths, and repeat them 5 times, which will make , and will take 2 squares and 2 spaces.

Ans. 24. 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 u, or three whole ones, which added to 12 make 15.

32. 2 times 3 are 6, and 2 times are , which added to 6 make 69.

40. 10 barrels of cider at 3 dollars and i a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times is 50, or 8 and 3 of a dollar. Ans. 382 dollars.

C. 2. i to each would be 3 times , or, which are 2 oranges.

3. or 2 bushels. 4. 7 times are , or 5 gallons. 5. 8 yards and for 2 yards, that is, 10 yards.

6. 4 times 2 are 3, and 4 times are **, or 2, which added to 8 make 103 bushels.

12. It would take 1 man 3 times as long as it would 3 men. Ans. 137 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. 38, rods 15. Ans. 12 yards.

SECTION X.

A. 21. of 1 is į: } of 2 is 2 times as much or ģof 4 is , or 15. of 5 is t, or 15. of 6 is h, or 2. of 7 is ž, or 25.

27. } of 1 is ž. of 2 is . } of 3 is . } of 7 is }, or lý.

This manner of reasoning may be applied to any number. To find 4 of 38: it is , for of 38 is 38 times as much as of 1, and 1 of 1 is –, consequentpy of 38 is *, and 4 is 5%.

40. To find of a number, į must be found first, and then will be 2 times as much. of 7 is }, and 2 times are , or 41.

74. į of 50 is 50, or 55 ; is 4 times as much; 4 times 5 are 20, 4 times are, or 27, which added to 20 make 224.

Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers.

B. 2. Ans. l; apiece. 3. į of 3 is ; of a bushel apiece. 4. of 7 is 4}; he gave away 41 and kept 2. 6. 1 half dollar a yaxd, or 50 cents.

7. } of 7 is ž, or lz; of a dollar is of of 100 cents, which is 40 cents. Ans. I dollar and 40 cents a bushel.

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

9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and ș, and 2 bushels will cost 5f dollars. Ans. 5 dollars and 2 shillings, or 33% cents.

13. If7 pounds cost 40 cents, I will cost 54 cents; 10 pounds will cost 574 cents.

16. 1 cock would empty it in 6 hours, and 7 coeke

would empty it in 4 of 6 hours, or 4 of 1 hour, which is of 60 minutes; of 60 minutes; is 513 minutes.

SECTION XI.

A. 2. 2 halves of a number make the number ; consequently 1 and 1 half is the half of 2 times i and 1 half, which is 3.

15. 4 is į of 5 times 4 and {, which is 221.
17. 41 is of 9 times 4, which is 39%.
B. 4. 5 is 3 times į of 5, which is , or if.

30. If 8 is of some number, į of 8 is of the same number. of 8 is 25, 2 is of 4 times 2, which is 10%; therefore 8 is of 103.

40. If 8 is, of 8 ist; of 8 is, is of , or 9%; therefore 8 is of 9 .

52. If of a ton cost 23 dollars, į of a ton must be į of 23, that is 43 dollars, and the whole would cost 9 times as much, that is, 413.

69. 1 of 65 is 7 ; 7 is į of 5 times 7%, which is 364. 65 is f of 36.

C. 4. 37 is & of 32, which taken from 37 leaves 4. Ans. 44 dollars. 5. 7 feet must be { of the whole pole.

6. If he lost , he must have sold it for jof what it cost. 47-is of 604. Ans. 60 dollars and 428 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. 831 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 1 ; 5 of 60 minutes are 46 minutes and of; of 60 seconds are 40 seconds. Ans. 10 hours, 46 minutes, 40 seconds.

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

13. 5 is į 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 1l yards that is 3 quarters wide.

15. of 37 dollars.
16. as much

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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 a how many Å? expressed in words, is, in 7 how many sixths ? Ans. 4.

14. Reduce 81 to an improper fraction; that is, in 8 and 3 tenths, how many tenths ? Ans. 13.

B. 8. 23 are how many times 1 ? That is, in 23 sevenths how many whole ones ? Ans. 3.

IX. B. 3. How much is 5 times 64 ? That is, how much is 5 times 6 and 4 sevenths ? Ans. 324.

V. & X. 15. What is i of 27 ? That is, what is 5 eighths of 27 ? Ans. 16%.

VI. & XI. A. 8. 79 is of what number? That ix, 7 and 6 sevenths is 1 eighth of what number?

Ans. 624.

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 fisths of what number ? Ans. 69.

Explanation of Plate III.

Plate III is intended to represent fractions of unity, divided into other fractions ; it is, therefore, an extension of plate II. It differs from it, only in this, that besides the vertical divisions, the squares are divided horizontally, so as to cut the fractions of the square into fractions of fractions. 'The horizontal lines are dotted, but they are to be considered as lines.

This plate, like the preceding, is divided into ten rows of squares, each row containing ten equal squares. In the first row, the first square is undivided, the 9 following squares are divided by horizontal lines into from two to ten equal parts. In all the other squares the vertical divisions are the same as in Plate II, and besides this, each row is divided horizontally in the same manner as the first row.

By means of this double division, the 2d row presents a series of fractions, from halves to twentieths. The 3d row presents a series from thirds to thirtieths, and so on to the 10th row, which presents a series from tenths to hundredths.

The 2d row, besides presenting halves, fourths, sixths, eighths, &c. shows also halves of halves, thirds of halves, fourths of halves, &c. and shows their ratios with unity.

The 3d row, besides thirds, sixths, ninths, &c. shows halves of thirds, thirds of thirds, &c. and their ratios with unity. The other rowi proport analagoun divisions.

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