CHAPTER IX. COMMON FRACTIONS. Art. 116. A fraction is a number expressing one or more of the equal parts of a unit. In reference to the number of parts into which the unit is divided, and to the style of writing them, fractions are of two kinds, namely common, and decimal. A common fraction is used to express any part or parts of a unit, such as halves, thirds, fourths, tenths, hundredths, &c. A decimal fraction is used to express decimal parts of a unit, such as tenths, hundredths, thousandths, &c. NOTE.-Common fractions are often called vulgar fractions. NOTATION OF COMMON FRACTIONS. Art. 117. A common fraction is expressed by writing one number above another with a line between them. Thus, two-fifths is written, three-eighths, six-tenths, &c. The denominator of a fraction is the number below the line. It is so called because it denominates, or names, the parts of the unit. Thus, in the lower number, 5, represents the division of the unit into five equal parts, and the fraction expresses fifths. The numerator of a fraction is the number above the line. It is so called because it numerates, or numbers, the parts expressed by the fraction. Thus, in the upper number, 2, denotes that two of the five-fifths of a unit are expressed by the fraction. Hence the numerator is a cardinal number, and the denominator an ordinal adjective, viewed either as a noun, or as having the noun part or parts understood. The terms of a fraction are its numerator and denominator. They are so named when alluded to in general language, such as, the upper term, the lower term, both terms, &c. Art. 118. To write a common fraction. Rule.— Write the numerator; under it draw a horizontal line, and under the line write the denominator. NOTE.-In business, the line between the terms of a fraction is sometimes drawn downwards obliquely to the left, thus, 3⁄4, &c. EXAMPLES FOR PRACTICE. Write in figures, 1. Three-eighths; three-sixteenths; three-twelfths. 4. Seven-twelfths; eight-forty-firsts. 5. Ten-seventeenths; nine-fortieths. 6. Three-fourths; four-fifths; six-sevenths. 7. Three-elevenths; seven-sixteenths. 8. Thirteen-fifty-firsts; twenty-sixty-firsts. 9. Twenty-five-sixty-thirds; sixty-ninetieths. 10. Seventy-one-one-hundred-and-seconds. 11. One-hundred-and-five-two-hundred-and-thirds. NUMERATION OF COMMON FRACTIONS. Art. 119. To read a common fraction. Rule.-Pronounce the numerator as a whole number, and then the denominator as an ordinal number. NOTE. When the denominator is 2, it is read as half or halves, and, when it is 4, it is read as quarters, or fourths. 31 73 16 .. : : 19 130 8. Read 11: 21: 70% : 4318:1:8:27:27 : 83: : 8 48 108 7508 9 65 10. Read2: 1:11:103:2202:13:13:38:85: 14 SIGNIFICATION OF FRACTIONS. Art. 120. A common fraction may have two meanings, viz.: FIRST.-The division of a unit into equal parts, and the expression of a certain number of these parts. Thus, 2 over 5, or , may represent the division of a unit into 5 equal parts, called fifths, and may express two of these fifths, that is, twofifths of 1. SECOND. The division of the numerator into as many equal parts as there are units in the denominator, and the expression of one of those parts. Thus, may represent the division of 2 into 5 equal parts, and may express one of those parts, that is one-fifth of 2. VALUE OF FRACTIONS. Art. 121. The value of a fraction is the quantity which it expresses, and is the quotient of the numerator divided by the denominator. Thus, the value of § is 2. DEMONSTRATION. Since 3 thirds equal 1 unit, 6 thirds are as many units as 3 thirds are contained times in 6 thirds. Since 3 thirds are contained in 6 thirds 2 times, 6 thirds are equal to 2 units. Both meanings of a fraction have the same value. Thus, twofifths of 1 is the same value as one-fifth of 2. This appears from the following demonstration and illustration: DEMONSTRATION.-Since one is 5 fifths of 1, 2 if equal to 2 times 5 fifths of 1, that is 10 fifths of 1, and 1 fifth of 2 must equal 1 fifth of 10 fifths of 1, that is, 2 fifths of 1. ILLUSTRATION. If two equal lines are each divided into 5 equal pieces, 2 of these pieces are two-fifths of one line, and, also, one-fifth of the whole ten pieces, into which the two lines were cut. The value of a fraction does not depend upon the value of the terms, but upon the number of times that the lower term is contained in the upper. Thus, I, I, I, 4, 4, 43, &c. have the same value, though the terms have different values. 18 VARIATIONS OF THE VALUES OF FRACTIONS. Art. 122. Since the value of a fraction is the quotient of the numerator divided by the denominator, it varies with the relative variations of the terms. (See Art. 79.) Preposition I.-The value of a fraction varies directly as the numerator. DEMONSTRATION.-The same denominator is contained in twice the numerator, twice as many times; in three times the numerator, three times as many times; in one-half the numerator, one-half as many times; and one-third of the numerator, one-third as many times, &c. = = 6, we ob ILLUSTRATION.-If we double the numerator in 24 tain 48 twice 6, or 12. If we divide the numerator by 2, we obtain one-half of 6, or 3; &c. Hence, multiplying the numerator by a number multiplies the value of the fraction by that number; and dividing the numerator by a number divides the value of the fraction by that number. Preposition II.-The value of a fraction varies inversely as the denominator. DEMONSTRATION.—If the numerator is constant, twice the denominator is contained in it one-half as many times; three times |