268973 30. Of 2371; + 49617 + 84928 t. 59118 + 43818 ? 31. Off-? 32. Of 13 - 18? 33. Of 83257 494_6 %, ? 268943 34. Of 2475 - 1933 ? NOTE. Compare the 34th with the following:- What is the valus of 24 qr. 7 lb. — 19 qr. 21 lb. By writing the work of the 34th example in the following form, and placing opposite it the work of the example in this note, the resemblance of the two is very clearly exhibited. Ones. Twenty-fifths. Qr. Ib. 2425 = 24 7 7 193} 19 21 19 4}} = 4 11 = Ans. 24 21 11 = Ans. What is the value 35. Of 2471- 1591%? 129. One Number a Fractional Part of another. What part of 4 is 1 ? Answer. 1 is one fourth of 4, because ~ times 1 are 4.* What part1. Of 6 is 1 ? 3. Of 9 is 1 ? 5. Of 16 is 1 ? 2. Of 2 is 1 ? 4. Of 12 is 1 ? 6. Of 243 is 1 ? 7. What part of 8 is 3 ? Solution. — Since 1 ☆ of 8, 3 must equal ß of 8. What part 8. Of 9 is 4 ? 10. Of 3 is 2 ? 12. Of 6 is 5 ? 9. Of 11 is 7 ? 11. Of 8 is 5 ? 13. Of 13 is 4 i | * i. e., Because 1 taken 4 times will equal 4 49. Of 11 3 is 1 lb. ? I 50. Of 8 da. is 1 w.? 51. What part of & is 1 ? Solution. 1=6, and is the same part of that 6 is of 5, which is f. Hence, 1 of &, or 1} times What part 52. Of is is 1 ? 57. Of 33 is 1 ? 53. Of is 1 ? 58. Of 4 is 1 ? 54. Of 4 is 1 ? 59. Of 34 is 1 ? 55. Of is 1 ? 60. Of is 1 ? 56. Of is 1 ? 61. Of 162 is 1 ? 62. What part of 13 is 25 ? Solution. iš= }; 2g :*?; and * = 1 of 14 =14 times 63. What part of 1 m. 3 fur. is 2 m. 5 fur. ? Solution. 1 m. 3 fur. = 11 fur.; and 2 m. 5 fur. = 21 fur.; 21 fur. it of 11 fur., or 111 times 11 fur. What part 64. Of 1 d. 3 qr. is 11 d. 1 qr.? 71. Of 15 pk. is 3 pk. 7 qt.? Suggestion. — Reduce i lb. and also 3 oz. 5 dwt. 17; gr. to the lowest denomination mentioned, i. e., to fifths of a grain. What part 73. Of 1 gal. is 3 qt. 1 pt. 3 gi.? 130. Other Methods of expressing the Value of a Fraction. (a.) 1. Explain the fraction. Answer. = 3 times Ā, as i of a number = 3 times of the number. In the same way explain the meaning – 8. Of .4 9. Of .007 4. Of g. 7. Of .15 10. Of .0346 (6.) Explain each of the above by the following method. , or of 1,= 1 of 3, as of a number 1 of 3 times the num ber. For if each of 3 equal things should be divided into 4 equal parts, and 1 of the parts of each taken, the result would equal 1 of 1. NOTE. This may be illustrated to the eye, by taking 3 equal lines, and dividing them into 4 equal parts, arranged thus one of the parts will then contain 4 of 3 lines, which, as will be seen, is equal to of a line. 11. To what simple fraction is of 3 equal ? Answer. 1 of 3 of 1= . (c.) To what simple fraction is each of the following equal ? 12. of 2. 16. f of 11. 20. t of 832. 13. 4 of 4. 17. 1 of 4. 21. 7 of 98. 14. ii of 6. 18. i's of 15. 22. z of 493. 15. of 7. 19. of 49. 23. 7 of 861. 131. To find a Fractional Part of a Number. 1. What is į of 5459 ? First Solution. šof 5459 7 times f of 5459 ; of 5459, found by dividing by 8, is 6823, and 7 times 6823 : 4776. Second Solution. šof 5459 = 3 of 7 times 5459 ; 7 times 5459 = 38213, and f of 38213 = = 4776. a = The following forms of writing the work exhibit very clearıy the reason for every step which is taken. The letters a, b, &c., are used as explained in the note on the 88th page. First Form. Second Form. %= 5459 5459 f of a = b= 6823 7 Xa=b 7 x b= } of a = 4776 $ of b 4776 NOTE. - It will be seen that by both methods of solution, we multiply by the numerator, 7, and divide by the denominator, 8. The only difference between them is, that by the first we divide before we multiply, while by the second we multiply before we divide. The first method involves somewhat smaller numbers than the second. What is the value of each of the following ? 2. of 3843. 4. of 7148. 6. 63 of 46935. 3. šof 7987. 5. 1 of 58437. . 7. 17 of 87516. 8. What is the value of .0014 of 16.427 ? First Solution. .0014 of 16.427 = 14 times .0001 of 16.427; .0001 of 16.427, found by removing the point four places towards the left, is 0016427, and 14 times this is .0229978 = Ans. The work may be written in full after either of the following models First Model. a = 16.427 Second Model. 10000 ) 16.427 .0016427 14 .0229978 = Ans. Second Solution. .0014 of 16.427 = .0001 of 14 times 16.427 ; 14 times 16.427 = 229.978, and .0001 of 229.978, found by removing the point four places towards the left, is .0229978 = Ans. The work may be written in full after either of the following models First Model. a = 16.427 |