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 ? Ans. Thirds. Why? Because, the more parts a thing is divided into the smaller the parts must be. squares and 1 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 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. į signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times Į third is 2 parts, or . 6. 7 times is 1, or 25. 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 y, 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 *, or three whole ones, which added to 12 make 15. 32. 2 times 3 are 6, and 2 times are y, which added to 6 make 64. 40. 10 barrels of cider at 3 dollars and í a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times is 5, or 8 and 2 of a dollar. Ans. 389 dollars. C: 2. to each would be 3 times , or, which are 21 oranges. 3. or 2 bushels. 4. 7 times are 4, or 51 gallons. 5. 8 yards and or 2 yards, that is, 10 yards. 6. 4 times 2 are 3, and 4 times are 4, or 23, which added to 8 make 103 bushels. 12. It would take 1 man 3 times as long as it would 3 men. Ans. 132 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 iu 1 day. Ans. 387 rods. ,15. Ans. 12 yards. SECTION X. A. 21. of 1 is į of 2 is 2 times as much, or of 4 is f, or 11° į of 5 is , or 11. of 6 is , or 2. į of 7 is 1, or 21. 27. į of 1 is t. of 2 is z. of 3 is . of 7 is }, or 15. This manner of reasoning may be applied to any number. To find of 38: it is 48, for 4 of 38 is 38 times as much as of 1, and 1 of 1 is , consequently of 38 is y, and is 57. 40. To find of a number, { must be found first, and then will be 2 times as much. of 7 is }, and 2 times are \, or 44. 74. of 50 is 5, or 51 ; is 4 times as much ; 4 times 5 are 20, 4 times are, or 24, which added to 20 make 227. 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 43; he gave away 44 and kept 2 6. i half dollar a yard, or 50 cents. 7. } of 7 is }, or 14 ; of a dollar is of 100 cents, which is 40 cents. Ans. I dollar and 40 cents a bushel. 8. á of 8 is 13. 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, 1 bushel will cost 2 dollars and I, and 2 bushels will cost 5f dollars. Ans. 5 dollars and 2 shillings, or 33; cents. 13. If 7 pounds cost 40 cents, I will cost 54 cents; 10 pounds will cost 571 cents. 16. I 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 517 minutes SECTION XI. A. 2. 2 halves of a number make the number; consequently 1 and 1 half is the balf of 2 times i and 1 half, which is 3. 15. 4 is of 5 times 4 and {, which is 224. 30. If 8 is of some number, į of 8 is of the same number. Í of 8 is 21, 2 is of 4 times 2, which is 10%; therefore 8 is of 10%. 40. If 8 is, of 8 ist; of 8 is, is of 49, or 9%; therefore 8 is of 9 52. If of a ton cost 23 dollars, ; of a ton must be } of 23, that is 4 dollars, and the whole would cost 9 times as much, that is, 41. 69. of 65 is 74; 74 is į of 5 times 7%, which is 36. 65 is g of 364. C. 4. 37 is of 32%, which taken from 37 leaves 4. Ans. 41 dollars. 5. 7 feet must be s 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 429 cents. 1 Miscellaneous Examples. 1. The shadow of the staff is of the length of the staff; therefore th> shadow of the pole is of the length of the pole. 67 is f' of 83. Ans. 831 feet, 2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7 ; 1 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. 3. 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 1lf yards that is 3 quarters wide. 15. of 37 dollars. 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 f? expressed in words, is, in 7 how many sixths ? Ans. . 14. Reduce 8j to an improper fraction ; that is, in 8 and 3 tenths, how many tenths ? Ans. I... B. 8. 23 are how many timos 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. 324. V. & X. 15. What is of 27 ? That is, what is 5 eighths of 27 ? Ans. 167. VI. & XI. A. 8. 79 is į of what number ? That is, 7 and 6 sevenths is 1 eighth of what number? Ans. 624. |