Fractions. 5. Any fraction that is neither compound nor complex is a simple fraction. Ž, Ji, f4, are simple fractions. 6. A fraction whose denominator is 1 with one or more zeros annexed to it, is a decimal fraction. o, .7, .25, 76, are decimal fractions. NOTE 1.—The denominator of a decimal fraction may be 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.-A11 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}, .254, .0563, are complex decimals. 177. There are three aspects in which fractions should now be considered. 1. 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 (fractional units) taken. Fractions. II. THE DIVISION ASPECT. The numerator of a fraction is a dividend; the denominator, a divisor, and the fraction itself, a quotient : thus, in the fraction s, the dividend is 5; the divisor, 8, and the quotient, s. NOTE.—In the case of an improper fraction, as q, 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 75, 7 is the antecedent, 10 the consequent, and fg the ratio. NOTE 1.—This relation may be more readily seen by the pupil in the case of an improper fraction. In the fraction 42, 12 is the antecedent; 4, the consequent; 4, 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. 8 (units of measurement) is the antecedent; 1 (unit of measurement) is the consequent, and the pure number 8 is the ratio. 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. Fractions. 3. Reduce jo i to its lowest terms. 200 Operation. Explanation. Dividing each term of 188 by 10 we have 1 fourth as many parts, which are 4 times as large. Hence, 188 = 4. 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. (a) Find the sum of the ten results. I 4. Reduce to higher terms — to 120ths. Operation. 120 8 = 15. Explanation. In 15. there are 15 times as many parts as there are in , and the parts are 1 fifteenth as large. Hence, in 75 5 x 15 8 x 15 120 * Divide each term by 12. This involves the reduction of a complex to a simple fraction; but it will lead to thoughtful work for the pupil to solve such problems in this manner. + Divide each term by 1. # If the pupil has not had sufficient practice in addition of fractions to do this. the finding of the sum may be omitted until the book is reviewed. Fractions. 37 9 20 Reduce to higher terms to 160ths. (3) (7) PE (8) 37 (9) 33 (10) } (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 ge 24 27 36 36 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 ft, do, and f7 to equivalent fractions having their 'least common denominator. OPERATION. 11 X 4 44 (2) 120 - 30 = 4 30 x 4 120 9 X 3 27 (3) 120 + 40 = 3 40 x 3 120 37 x 2 74 (4) 120 - 60 2 60 x 2 120 Fractions. Reduce to equivalent fractions having their 1. c. d. (1) 15 and 2 (6) +, %, and 11 (2) 3% and 18 (7) 29, , and zo (3) {; and 11 (8) 3%, , and (4) 37 and (9) $8, , and tobe (10) 17, Yo, and 5 6 179. To add common fractions. RULE.- Reduce the fractions if necessary to equivalent fractions having a common denominator, add their numerators, and write their sum over the common denominator. EXAMPLE. Add 11, 17, and 8. NOTE.-If the work that precedes this article has been well done, no explanation of the foregoing will be necessary. Pupils have already learned (presumably before using this book) (1) that fractions may be reduced to higher terms, (2) that two or more fractions whose denominators are not alike may be reduced to higher terms with like denominators, (3) that a common denominator of two or more fractions with unlike denominators, is a common multiple of the given denominators, and (4) that in reducing a fraction to higher terms the numerator and denominator must be multiplied by the same number. The simple problem of adding 44 180ths, 102 180ths, and 159 180ths, is not unlike the problem of adding 44 apples, 102 apples, and 159 apples. (For a continuation of this work, see page 91.) * This work may be omitted until the subject of fractions is reviewed. |