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an entire unit, or whole one. 2d, explain the divisions into two, three, four, &c. parts. 3d, teach him. to name the different parts. Make him observe that the name shows into how many parts one is divided and how many parts are taken, in the same manne as it does when applied to larger numbers. # for example, shows that one thing is to be divided into 7 equal parts, and 4 of those parts are to be taken. 4th, make the pupil compare the different parts together, and observe which is the largest. Ask him such questions as the following: Which are the smallest halves or thirds 1 Ans. Thirds. Why? Because, the more parts a thing is divided into, the smaller the parts must be.

A. 15. On plate II., count two squares in the second row, and then ascertain the number of spaces or halves in them. There are 4 halves.

21. In the 2d row take 3 squares and 1 space in . the 4th square ; then count the spaces. Ans. 7. halves.

37. In the 3d row take 5 squares, and 2 spaces in the 6th; then count the spaces or thirds. Ans. 17 thirds.

54. In the 5th row take 6 squares, and 4 spaces in the 7th square; then count the spaces or fifths. Ans. 34 fifths.

B. 2. 'This operation is the reverse of the last. In the 2d row count 4 spaces or halves, and see how many squares or whole ones it takes. It will take 2.

38. In the 9th row count 48 spaces or 9ths, and . see how many squares or whole ones it takes. It will take 5 squares and 3 spaces in the 6th. Ans. 5 whole ones and #.

SECTION IX.

A., 2, # signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times I third is 2 parts, or 4.

6. 7 times ; is 4, or 24.

10. On the plate in the 3d row, 5 times 3 are o', 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. 23.

B. 4. 4 times 2 are S, 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 of, 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 64 40. 10 barrels of cider at 3 dollars and # a barrel ; 10 barrels at 3 dollars, would be 30 dollars, then 10 times ; is o', or 8 and 3 of a dollar. Ans.

38; dollars.

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which added to 8 make 104 bushels. * - 12. It would take 1 man 3 times as long as it would 3 men. Ans. 133 days. 14. 3 men would build 3 times as much as I man ; and in 4 days they would build 4 times as much as in 1 day. Ans. 384 rods. 15. Ans. 12 yards.

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

B. 2. Ans. 14 apiece. 3. 4 of 3 is ; ; ; of a bushel apiece. 4. # of 7 is 44; he gave away 44 and kept 24 6. 1 half dollar a yard, or 50 cents. 7. 4 of 7 is 4, or 13; # of a dollar is 4 of 100 cents, which is 40 cents. Ans. I dollar and 40 cents a bushel. 8. of 8 is 1}. 3 of 100 is 333. 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 4, and 2 bushels will cost 54 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

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1. The shadow of the staff is 4 of the length of the staff; therefore the shadow of the pole is 4 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 g; ; of 60 minutes are 46 minutes and #; # of 60 seconds are 40 seconds. Ans. 10 hours, 46 minutes, 40 seconds. 10. Find 4 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 114 yards that is 3 quarters wide. 15. 3 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 7 how many # 1 expressed in words, is, in 7 how many sixths 1 Ans. *. 14. Reduce 8;or to an improper fraction; that is, in 8 and 3 tenths, how many tenths Ans. }}. B. S. ** are how many times 1 1 That is, in 23 sevenths how many whole ones 1 Ans. 34. IX. B. 3. How much is 5 times 64 That is, how much is 5 times 6 and 4 sevenths 1 Ans. 324. W. & X. 15. What is # of 27 ? That is, what is 5 eighths of 27 ? Ans. 16}. VI. & XI. A. 8. 7# is ; of what number 1 That is, 7 and 6 sevenths is 1 eighth of what number 3 Ans. 62;.

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