FRACTIONS GENERAL DEFINITIONS 70. Any one of the equal parts of a unit is called a fractional unit. Thus, one half, one fifth, and one tenth are fractional units. 71. Any number of equal fractional units is a fraction. Thus, one third, four fifths, five fifths, and eight fifths are fractions. 72. The number of equal parts into which a unit is divided to form fractional units is the denominator, and the number of equal fractional units taken to make a fraction is the numerator, of the fraction. 73. The numerator and denominator are together called the terms of the fraction. 74. A fraction is usually written in figures by writing the denominator below, and the numerator above, a line. Thus, four fifths is written 75. A fraction whose numerator is less than its denominator; that is, a fraction whose value is less than unity, is called a proper fraction; as, ૐ. 76. A fraction whose numerator is equal to or greater than its denominator; that is, a fraction whose value is equal to or greater than unity, is called an improper fraction; as,, . 77. A number made up of a whole number and a fraction combined is called a mixed number. Thus, 21, which equals 2+, is a mixed number. 78. A whole number is called an integral number or an integer. 79. A fraction may be considered to denote that the numerator is to be divided by the denominator. 80. The inverse of a given fraction is the fraction obtained by interchanging the terms of the given fraction; as, is the inverse of . 81. The reciprocal of a number is one divided by the number. Thus, the reciprocal of 5 is §. GENERAL PRINCIPLES 82. Multiplying the numerator of a fraction multiplies the value of the fraction. Thus, 3 x = 3 x 2 org, because if the numerator is multiplied by 3 and the denominator remains unchanged, there will be 3 times as many fractional units, each of the same value as before. 83. Multiplying the denominator of a fraction divides the value of the fraction. Thus, ÷ 3 = 2 or, because if the denominator is multi plied by 3 and the numerator remains unchanged, there will be the same number of fractional units as before, but each will be only one third as great. 84. Dividing the numerator of a fraction divides the value of the fraction. Thus, ÷ 3 = 9÷3 or, because if the numerator is divided by 3 and the denominator remains unchanged, there will be only one third as many fractional units, each of the same value as before. 85. Dividing the denominator of a fraction multiplies the value of the fraction. 7 Thus, 3×12 = 12÷3' or, because if the denominator is divided by 3 and the numerator remains unchanged, there will be the same number of fractional units as before, but each will be three times as great. 86. Multiplying both terms of a fraction by the same number does not alter the value of the fraction. 3 x 4 Thus, = or, because multiplying the numerator by 3 multiplies the value of the fraction by 3, and multiplying the denominator by 3 divides the value of the fraction by 3, and doing both, multiplies and divides the value of the fraction by 3, which does not alter its value. 87. Dividing both terms of a fraction by the same number does not alter the value of the fraction. Thus, 12 ÷ 4 16÷4' or, because dividing the numerator by 4 divides the value of the fraction by 4, and dividing the denominator by 4 multiplies the value of the fraction by 4, and doing both, divides and multiplies the value of the fraction by 4, which does not alter its value. Exercise 19 1. Name the fraction whose numerator is 4 and denominator 5. 2. Name the fraction whose numerator is 7 and denominator 4. 3. Name the fraction whose numerator is 5 and denominator 5. 4. Name the number made up of 3 and combined. 5. Name three proper fractions each having the denominator 4. 6. Name three improper fractions each having the numerator 4. 7. What is the inverse of ? 8. What is the inverse of? 9. Express as a fraction, 3÷2; 4 ÷ 5; 3 ÷ 3; 10÷9. REDUCTION 88. Reduction is the process of changing the form of an expression without changing its value. 89. A fraction is said to be in its lowest terms when its numerator and denominator are prime to each other. Thus, when reduced to its lowest terms equals 90. A fraction may be reduced to its lowest terms by dividing both of its terms by their greatest common divisor. 91. Fractions are said to be similar when they have a common denominator. Thus, and are similar fractions. 92. Fractions having different denominators may be reduced to fractions having a common denominator equal to the least common multiple of their denominators. Thus,,, and may each be reduced to 60ths. NOTE 1. Any common multiple of the denominators of two or more fractions may be taken as a common denominator of them. Thus, 30, 60, 90, etc., may each be taken as a common denominator of,, and §. NOTE 2. If fractions having different denominators are in their lowest terms, the least common multiple of their denominators is their least common denominator (1. c. d.). 93. A mixed number is equal to the whole number multiplied by the denominator of the fraction, plus the numerator of the fraction, all divided by the denominator of the fraction. Thus, 183= 5 × 18+ 3 |