CHAPTER X COMMON FRACTIONS 183. Fraction. A number that shows what part or what number of parts of a unit is taken is called a fraction. Fractions have already been studied in the Fourth Year. They are now reviewed and are studied more carefully. 184. Common Fraction. A fraction expressed by two numbers, one under the other with a line between them, is called a common fraction. 185. Terms of a Fraction. In a common fraction the number above the line is called the numerator ; the number below the line is called the denominator ; the two together are called the terms of the fraction. In the fraction of the numerator is 3 and the denominator is 4. The denominator (namer) shows into how many equal parts the unit has been divided, and the numerator (numberer) shows how many of these equal parts have been taken. 186. Unit Fraction. A fraction whose numerator is one is called a unit fraction. Thus į is a unit fraction. It is also called a fractional unit. 187. Proper Fraction. A fraction whose numerator is less than the denominator is called a proper fraction. For example, i is a proper fraction. 188. Improper Fraction. A fraction whose numerator is not less than the denominator is called an improper fraction. For example, and į are improper fractions. 189. Fraction as an Expressed Division. In this figure we see that f of 1 inch equals of 2 inches, or the result of of 1 inch 1 of 2 inches dividing 2 inches by 3; that is, is the same as } of 2, or 2 divided by 3. Therefore, A fraction is an expressed division, the numerator being the dividend and the denominator the divisor. 190. Reduction of a Fraction to Lowest Terms. Since we may divide both terms of a fraction by the same number without changing the value of the fraction, we may continue this division until the terms are prime to each other. If the terms of a fraction are prime to each other, the fraction is expressed in lowest terms. For example, 19 is expressed in lowest terms, but is not. Therefore, to reduce a fraction to lowest terms, cancel all the factors common to both numerator and denominator. For example, reduce jf to lowest terms. 191. Reduction of a Fraction to Higher Terms. By multiplying both terms of a fraction by the same number we may reduce a fraction to higher terms. For example, reduce U to sixtieths. Here 60 + 12 = 5, so that the denominator must be multiplied by 5 to make 60. Therefore we multiply both terms by 5, and 11 = . Therefore, to reduce a fraction to a fraction having a higher denominator, divide the required denominator by the given denominator and multiply both terms of the fraction by the quotient. 192. Reduction of an Integer to an Improper Fraction. An integer may be expressed as an improper fraction with any required denominator. Reduce 3 to an improper fraction with denominator 7. We may simply think of 3 as equal to i. We have, then, the same case as in § 191. Multiplyiug both terms by 7, we have { = 24. Therefore, to reduce an integer to a fraction with a given denominator, multiply it by the given denominator and under this product write the denominator. 193. Mixed Number. A number composed of an integer and a fraction taken together is called a mixed number. Thus 2 is a mixed number. It is read "two and three fourths." 194. Reduction of an Improper Fraction. An improper fraction may always be reduced to an integer or a mixed number. For example, required to reduce it to an integer or a mixed number. Dividing both terms by 12, we have f, which equals 6. Since is is an expressed division, we divide 36 by 16 and the quotient is 216, or 21. Therefore, to reduce an improper fraction to an integer or a mixed number, divide the numerator by the denominator. EXERCISE 101 Reduce to integers or mixed numbers : 96 96 39. 480 96 22. 73 4. 48. 5. 13. 28. 84 37. 38. 30. 448 3. 4. 12. 4 31. 99 40.6 13. 88 32. 94. 41. 458 14. 1%. 23. 6 42. 178. 24. 86 33.95 15. 9. 5. 6. 1 34. 4. 43. 138 25. 3.50 7. 48. 16. 46. 35.28 44. 168 26. 4. 8. 12. 17. 13 27. 36. 1982 9. 18 18. 46. If the numerator of a fraction is equal to the denom 56 45. 750 100 4 8. inator, to what integer is the fraction equal ? 47. If the numerator of a fraction is 24 times the denominator, to what mixed number is the fraction equal ? To what integer is it equal if the denominator is of the numerator? 195. Reduction of a Mixed Number. A mixed number may always be reduced to an improper fraction. For example, required to reduce 24 to an improper fraction. Since 1 equals 4 fourths (4), 2 equals 2 x 4 fourths, or 8 fourths (f); and 8 fourths and 3 fourths are 11 fourths (4). Therefore, to reduce a mixed number to an improper fraction, multiply the denominator by the integral part of the mixed number, and to the product add the numerator ; under this sum write the denominator. EXERCISE 102 Reduce to improper fractions : 11. 234 21. 836 1. 31. 31. 184 32. 54 33. 774 34. 334 35. 6% 1. 36. 1,0 37. 677 38. 53 39. 353 3. 40. 7241 10. 611 20. 211 41. How many eighths (3) are there in 3 ? in 3? in 37 ? 42. How many sixteenths (1) are there in ? in 5 ? in 53? 43. How many thirty-seconds (39) are there in 3' ? in 3? in 331? 44. How many tins, } of a pound each, can be filled from f of a pound of pepper? from 2 pounds ? from 23 pounds? |