COMMON FRACTIONS. 98. 1. If a unit, as an apple, is divided into two equal pieces, what part of the whole will one piece be? 2. If an apple is divided into three equal pieces, what part of the whole will one piece be? Two pieces? 3. If a single thing is divided into four equal pieces, what part of the whole will one piece be? Two pieces? Three pieces ? 4. How many halves in an apple? How many thirds? How many fourths? 5. What is meant by one half of a unit? By one third? By two thirds? By one fourth? By three fourths? 6. Which are the larger parts of an apple, halves or thirds? Thirds or fourths? 99. A Fraction is one or more of the equal parts of a unit. The Unit of the Fraction is the unit divided, and a Fractional Unit is one of the equal parts into which it is divided. 100. The Denominator of a fraction is the number that shows into how many equal parts the unit is divided. Thus, 'Three is the denominator of two thirds. 101. The Numerator of a fraction is the number that shows how many of the equal parts of the unit are taken. Thus, Two is the numerator of two thirds. 102. The Terms of a fraction are its numerator and denominator. Thus, 2 and 3 are the terms of the fraction. 103. A Common Fraction is a fraction expressed by writing the numerator above, and the denominator below, a dividing line. Thus, Three fourths of a dollar is written $2, 3 being the numerator, 4 the denominator, 1 dollar the unit of the fraction, and dollar the fractional unit. 104. An Integer may be expressed in a fractional form, by writing 1 under it for a denominator. Thus, 2 may be written, and read 2 ones; 7 may be written 1, and read 7 ones; etc. 105. A Proper Fraction is a fraction whose numerator is less than its denominator. Thus, and are proper fractions. 106. An Improper Fraction is a fraction whose numerator is not less than its denominator. Thus, § and 13 are improper fractions. 107. A Mixed Number is an integer and a fraction united. Thus, 31, read three and one fourth, is a mixed number. 108. A Fraction may be regarded as an indicated division (Art. 63), the numerator being the dividend, and the denominator the divisor. Thus, of 1 inch is the same as of 3 inches, or 3 inches divided by 4. 109. The Value of a fraction is the quotient of the numerator divided by the denominator. Express in figures: EXERCISES. 7. Three sevenths. 8. Seven elevenths. 9. Nine sixteenths. 10. Seventeen ones. 11. One twenty-first. 14. Three and three twenty-ninths. 12. Eleven thirty-seconds. 15. Twenty-three and three fifths. 13. Nineteen fortieths. 16. Eight and nine twelfths. REDUCTION OF FRACTIONS. ORAL EXERCISES. 17. In of an apple how many fourths of an apple? how many eighths? 18. In of an apple how many sixths? how many ninths? 19. Name a fraction equal to . Name a fraction equal to 3. 20. Express in terms 2 times as large. 3 times as large. 21. Change to 1%, to 1%, to3. 22. How is the fraction § changed to twelfths? 23. How many halves are there in ? How many thirds are there in1⁄2? 24. How many tenths of a melon in 25. Express in larger terms; smaller terms; 1 in smaller terms. 8? how many fifths? in larger terms; fin 110. Reduction of Fractions is changing their form without changing their value. equals 1 of it, the multiplication increasing the number of fractional units 4 times, and making each one fourth as large, so that the value of the fraction is not changed. 111. Principle. Multiplying both terms of a fraction by the same number does not change its value. Or, 27. Change to 12ths,,, 1, 3, 4, 8. 28. Change to 18ths,,,, 3, 8, 6, 3, §, 1. 29. Change to 20ths 1, 4, 1, 4, 1o, t, Po, fo, fo. 30. Change to 24ths 1, 3, 4, ↓, k, T2, 3, 3, 8, §, I, J, Ja, 11. 31. Change to 36ths 3, 4, 8, 5, 3, 11, 4, 3, P2, 12. 32. Change to 48ths 3, 3, 8, 8, 1, 12, 14, 1, 74, 16, 11. WRITTEN EXERCISES. 33. Change to twenty-fourths. 3 24 Solution. To change 8ths to 24ths, we must multiply both terms of the fraction by 3. Doing this, we obtain 24. Or, Since 1 is 24 twenty-fourths, } of 1 is of 24 twenty-fourths, or 3 twenty-fourths, and of 1 is 3 times 3 twenty-fourths, or 4. 34. Change to seventy-fifths. 112. To change a fraction to larger terms: Rule. Divide the required denominator by the given denominator, and multiply both terms of the fraction by the quotient. 113. A fraction is in its Smallest Terms when its terms have no common factor. equals of it, the division increasing the size of the fractional units 4 times, while their number is a fourth as large, so that the value of the fraction is not changed. 114. Principle. Dividing both terms of a fraction by the same number does not change its value. Change to smallest terms: 44. 4, t, i, †, fo, f2, 12, 12, 12, 12.. 15 45. 3, §, §, 12, 18, f‰, 13, 14, 18, 1§. 30 49., 18, 18, 19, 28, 45, 43, 19, 46, 38. 50. 18, 3, 88, 45, 18, 41, 83, 85, 85, 18. Solution. Dividing both terms of g by 3, we have 18, and dividing both terms of 18 by 5, we have, whose terms have no common factor; hence, changed to smallest terms is 3. |