FRACTIONS 160. A Fraction is a part of an integer. (a) Any quantity or value may be divided into any number of equal parts. One or more of these equal parts constitutes a fraction. (b) If a quantity be divided into four equal parts, each of them will be known as one-fourth (1) and three of them would be threefourths (). If the whole were divided into a thousand equal parts, each part would be called one one-thousandth (1000) and any number of these parts could be made to form the fractional value, as 748 thousandths (748). 161. A fraction is composed of two terms. One term shows the number of parts into which the original integer was divided. The other term shows the number of these equal parts taken to make the fraction. 162. Denominator is the name applied to the term showing into how many parts the integer was divided. 163. Numerator is the name applied to the term showing the number of these equal parts used in the fraction. 164. There are two methods of fractional notation, common fractions and decimal fractions. 165. A Common Fraction is expressed by writing the numerator above the denominator with a line between, thus, . 166. A Decimal Fraction is any fraction whose denominator is 10 or any multiple of 10. It is not necessary to write the denominators of decimal fractions. All that is required is to write the numerator and place it in the proper decimal column to show what denominator is intended, thus, six-tenths (1) may be written .6. Six-thousandths may be expressed by placing the figure six in the proper decimal column, thus, .006, and so on. 167. Every fractional value may be expressed either as a common fraction or as a decimal fraction. In some cases it is easier to use the common fraction form; while in others it is easier to use the decimal form expressing the same value. 168. A Proper Fraction is a common fraction whose numerator is smaller than its denominator. Its value is less than one. 169. An Improper Fraction is a common fraction whose numerator is larger than its denominator. Its value is more than one. 170. A Mixed Number is an integral quantity and a fraction taken together, as, 14. 171. In its decimal form this quantity would be called a Mixed Decimal and would be written 1.25. 172. It is not always possible to reduce a fractional value to an exact decimal. In such cases it is necessary to retain a common fraction as part of a decimal, thus, .163, read sixteen and two-thirds hundredths. 173. A Complex Decimal is a decimal whose right hand place is a common fraction. 174. A Complex Fraction is a common fraction, the right-hand place of whose numerator is itself a fraction, 175. Every common fraction represents an unperformed division. and also is the result of performing the operation 8÷9. It may be read "eight divided by 9” or it may be read “eight-ninths.” 2÷3 may each be read "2 divided by 3." 176. Reduction of Fractions is the process of changing the form of a fraction without changing its value. 177. Reduction of fractions comprises: Changing to lower terms. Changing to higher terms. Changing improper fractions to mixed numbers. CHANGE TO LOWER TERMS 178. Common fractions are in their lowest terms when the numerator and denominator are prime to each other; that is, when they have no common factors, no common divisor. 179. Decimal fractions, when written in the form of common fractions, may be reduced to lower terms when their numerator and denominator are not prime to each other. The decimal fraction, four-tenths, if written in its common fraction form, shows at once that its real or lowest value is because 2 is a common factor of both 4 and 10. In other words, if an integer is divided into ten parts and four of these parts are taken, the value of the fraction would be exactly the same as if the integer had been divided into five parts and two of these parts taken. 180. To Change to Lower Terms: Strike out all factors common to both the numerator and the denominator; or, divide both the numerator and the denominator by their G. C. D. ILLUSTRATED SOLUTIONS 420 181. Problem: Reduce 28 to lowest terms. Dividing Seven is a factor of both 420 and 525. the numerator, 420, by 7 gives a new numerator of 60. Dividing the denominator, 525, by 7 gives a new denominator of 75. Five is a common factor of both 60 and 75. Dividing 60 by 5 gives a new numerator of 12. Dividing 75 by 5 gives a new denominator of 15. Three is a common factor of 12 and 15. Divide 12 by 3 and we have a new numerator of 4. Dividing 15 by 3 gives a new denominator of 5. No factor is common to 4 and 5. Therefore, is the lowest equivalent of 525 Find the G. C. D. of 420 and 525 (pars. 155 and 156). The G. C. D. is 105. 4201054. Therefore, the new numerator is 4. 525÷105=5. Therefore, the new denominator is 5, making the fraction. = 5 9 27 9 This is a complex fraction. The first step is to change the numerator and the denominator to the same kind of parts, thirds. One unit=3; then 13; 9 units=27. Then the numerator equals 5 thirds and the denominator equals 27 thirds; 5 and 27 being prime to each other, 27 is in its lowest terms. Ans. . 27. = 183. Problem: Change 24 8 7 Proceeding as before, we find that the complex fraction is equal to 14. Both terms may be divided by 2, producing; 7 and 20 being prime to each other, is in its lowest terms. 20 NOTE. For practice problems in reducing fractions to lowest terms see par. 12. CHANGING TO HIGHER TERMS 184. In handling common fractions in business problems, it is sometimes necessary to change a common fraction from its lowest terms to an equivalent common fraction of some higher denomination. To do this it is necessary to introduce such factors into both the denominator and numerator as will change the value of the denominator from the given figure to that required. 185. To Change a Common Fraction to a Required Denominator: Divide the required denominator by |