FRACTIONS. 174. A fraction may be expressed by two numbers, one of them being written above and the other below a short horizontal line; thus, |, ±\, Itt 175. The number above the line is the numerator of the fraction; the number below the line, the denominator of the fraction. 176. Kinds Of Fractions. 1. A fraction whose numerator is less than its denominator is a proper fraction. i> f> 1t> are proper fractions. 2. A fraction whose numerator is equal to or greater than its denominator is an improper fraction. §, |, -2T7-, are improper fractions. Note.—The fraction .7 is a proper fraction. 2.7 may be regarded as an improper fraction or as a mixed number. If it is to be considered an improper fraction it should be read, 27 tenths; if a mixed number, 2 and 7 tenths. 3. Such expressions as the following are compound fractions: 3 0f 6 2 of > 5 nf 7 4. A fraction whose numerator or denominator is itself a fraction or a mixed number, is a complex fraction. Fractions. 5. Any fraction that is neither compound nor complex is a simple fraction. f> "if' 1^' are si^Qple fractions. 6. A fraction whose denominator is 1 with one or more zeros annexed to it is a decimal fraction. -j3¥, .7, .25, -^\, are decimal fractions. Note 1.—The denominator of a decimal fraction maybe expressed by figures, or it may be indicated by the position of the right-hand figure of its numerator with reference to the decimal point. When the denominator is thus indicated, the fraction is called a decimal, and is said to be written decimally. Note 2.—All fractions that are not decimal are called common fractions. A decimal fraction when not" " written decimally" (or thought of as written decimally) is usually classed as a common fraction. 7. A complex decimal is a decimal and a common fraction combined in one number. .7^, .25^, .056|, are complex decimals. 177. There are three aspects in which fractions should be considered. I. THE FRACTIONAL UNIT ASPECT. The numerator tells the number of things and the denominator indicates their name. In the fraction \ there are 5 things (magnitudes) called sevenths. In the fraction -| there are five fractional units, each of which is one eighth of some other unit called the unit of the fraction. Note.—The function of the denominator is to show the number of parts into which the unit of the fraction is divided; the function of the numerator, to show the number of parts taken. Fractions. The numerator of a fraction is a dividend, the denominator a divisor, and the fraction itself a quotient; thus, in the fraction f, the dividend is 5, the divisor 8, and the quotient f. Note.—In the case of an improper fraction, as f, it may be more readily seen by the pupil that the numerator is the dividend, the denominator the divisor, and the fraction (| = 2) the quotient; but the division relation is in every fraction, whether proper or improper, common or decimal, simple or complex. III. THE RATIO ASPECT.* The numerator of a fraction is an antecedent, the denominator a consequent, and the fraction itself a ratio; thus, in the fraction ^, 7 is the antecedent, 10 the consequent, and T7T the ratio. Note 1.—This relation may be more readily seen by the pupil in the ease of an improper fraction. In the fraction V> 12 is the antecedent, 4 the consequent, *£, or 3, the ratio. Note 2.—Every integral number as well as every fraction is a ratio. The number 8 is the ratio of a magnitude that is 8 times some unit of measurement to a magnitude that is 1 time the same unit of measurement. 178. Reduction Of Fractions. 1. The numerator and the denominator of a fraction are its terms. 2. A fraction is said to be in its lowest terms when its numerator and denominator are integral numbers that are prime to each other. * This may be omitted until the book is reviewed. Fractions. 3. Reduce £.$-$ to its lowest terms. Operation. Explanation. 1 n\160 _ 16 Dividing each term of \%$ by 10, we have '200 20' 1 tenth as many parts, which are 10 times as i g A large. Dividing each term of \% by 4, we have .*)j5q" = .=". 1 fourth as many parts, which are 4 times as large. Hence, J£{} = J. But 4 and 5 are prime to each other, and the fraction is in its lowest terms. Rule—Divide each term of the fraction by any common divisor except 1, and divide each term of the fraction thus obtained by any common divisor except 1, and so continue until the terms are prime to each other. Reduce to lowest terms: (1)55 (2)» (3) 3" (4)» v '375 v '650 v '270 v '340 (5) J* (6)H (7) — (8) — v '210 v '180 w 405 w 204 (9) ?!i* (io) 3+ v '100 v '4 (a) Find the sum of the ten results.^ 4. Reduce f to higher terms — to 120ths. Operation. Explanation. 120-1-8 = 15. In Jw there are 15 times as many parts as there are in §, and the parts are 1 fifteenth 5 X 15 _ ]_S_ as large. Hence, rfft = |. 8 x 15 120 * Divide each term by 12J. t Divide each term by }. } If the pupil has not had sufficient practice in addition of fractions to do thlgj the finding of the sum may be omitted until the book is reviewed. Fractions. Seduce to higher terms — to 160ths. (1)1 (2)tt (3) A (4)H (5) A (6)f (7) T% (8)H (»)H (10) W (a) Find the sum of the ten results. 5. Two or more fractions whose denominators are the same, are said to have a common denominator. 6. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having a common denominator. Example. ■| and £ may be changed to 12ths, 24ths, or 36ths. 2 _ 8— 2 — s _ 7 — ITS -f — 2T 7 — TB 1 — TB 7. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having their least common denominator. The 1. c. d. of two or more fractions is the 1. c. m. of the given denominators. Example. Change ^, -fa, and |^j- to equivalent fractions having their least common denominator. Operation. |