CHAPTER VI. FRACTIONS. SECTION I. Introduction and Definitions. 1. Let it be required to aistribute a dollars equally among b poor persons. What will be the share of each? a ANS. $. The number of dollars must be divided by the number of persons; but, as the division cannot be actually performed, all that can be done is, to represent it as above. So, too, if we suppose a = 3, and b = 5, we must indicate the answer in a similar manner; thus, . 3 These expressions,,, and others like them, arə called Fractions, from a Latin word, which signifie: broken, because the value of a fraction is always expressed in parts of a whole one. If, for instance, we cut an apple into 5 equal parts, and give 3 of those parts to a boy, he will have of the apple, which ex pression is read three fifths. in a similar manner; as, 2, one half;, two thirds; 2, three fourths;, five sixths; 3, seven eighths, &c. Other fractions are read The number below the line, which shows into how many parts the unit or quantity is divided, is called the Denominator. The number above the line, which shows how many of the parts are taken, is called the Numerator. a+b and, the nu merators are a, m, 6, 19, a + b, and 3; and the de nominators are b, n, 7, 4, x y, and z. A Proper fraction is one whose value is less than a unit; that is, whose numerator is less than its de is evident that the division here represented can be performed, either wholly or in part. Expressions consisting of a whole number and a raction, are called Mixed numbers; as,2,andb+; and 2 and 6 are called integers, or integral quantities. It often happens, both in Arithmetic and Algebra, that there is a remainder after division, which should be written above the divisor, and annexed to the quotient in the form of a fraction. Hence the origin of mixed numbers. [See Chap. V. Sec. III. and IV.] Since fractions always imply division, any quotient nay be expressed in the form of a fraction, the dividend bng the numerator, and the divisor the denominator As the value of any quantity is not altered when it is divided by a unit, we can convert an integer into a fraction by making 1 the denominator. Convert the following quantities into fractions: The numerator of a fraction being a dividend, and the denominator a divisor, it follows, that when these are equal to each other, as, the value of the fraction is 1; for 2 divided by 2 gives 1. Therefore a b and are all equal to each 1 2 3 4 5 a 2 3 4 5 a b other, the value of each fraction being unity, or 1. Hence, it is evident, that if both the numerator and denominator be either multiplied or divided by the same number or letter, the value of the fraction is not changed. On this principle are founded the rules for bringing fractions to a common denominator, and for reducing them to their least terms. 13. Express 6 in the form of a fraction, having 4 for its denominator. ANS. 24 4 It is evident that the numerator must be 4 times 6; for, when divided by 4, the denominator, the quotient must be 6. 14. Given the quantity x to be expressed in the form of a fraction, having y for a denominator. ANS. xy We may first convert a into this fraction, †, and then multiply both terms by y, according to the principles already given. In other words, to change a whole quantity to a fraction, having a given denominator, we multiply the whole quantity by the given denominator; and the product is the numerator of the fraction required. 15. Change 8 to a fraction, having 5 for its denominator. 16. Change a + b to a fraction, having c for its denominator. 17. Express x + y in the form of a fraction, having 5 z for its denominator. 18. Change a-5 to a fraction, having 2 +b for its denominator. 19. Change 5 abc to a fraction, whose denominator shall be 4 x. b 20. Express +65 in the form of a fraction, having 3 a z for the denominator. 21. Change 5 a for its denominator. 2+y to a fraction, having 6 22. Change 28-x+4y to a fraction, whose denominator shall be 2 a. 23. Express 2ab+x-5 in the form of a fraction, having 5 a b for its denominator. SECTION II. Reduction. It is often convenient, and sometimes necessary, to change the form in which a fraction or a mixed quantity is expressed. For instance, to prepare different fractions for addition or subtraction, we must always express them in other fractions, which shall have a common denominator. The process, by which the form of a fraction is changed, its value remaining the same, is called Reduction. As 5 fifths are equal to 1, 19 fifths must be equal To reduce an improper fraction to a whole or mixed quantity, Divide the numerator by the denominator, and annex the remainder, if any, to the quotient, in the form of a fraction. Reduce the following fractions to whole or mixed quantities: |