An entire quantity is one which has no fractional part as 3, 11, orb. An entire quantity, which is also a number, is called an integer. For example, 3 and 11 are integers. A mixed quantity consists of an entire quantity and a fraction. Thus, 51, 37, and a + 3 8 b d are mixed quantities. To reduce a mixed quantity to an improper fraction, multiply the entire quantity by the denominator of the fraction, add to this product the numerator of the fraction and place their sum over the denominator. To reduce 3 7 8 to an improper fraction, reduce 3 units to eighths by multiplying 3 by 8 obtaining 24; add this to 7 and the result, 31, is the numerator 7 31 of the fraction. 3 In the same way.a + b d ad + b d 8 8 To reduce an improper fraction to a mixed quantity divide the numerator by the denominator, write the quotient as the entire quantity of the mixed quantity and the remainder placed over the divisor (denominator of improper fraction) as the fraction. Examples. 1. Change 31 21 to a mixed number. 1 quotient 21 10 remainder SOLUTION. Divide the numerator 31 by the denominator 21, and place the remainder 10 over the divisor 21 as the fraction of the mixed quantity. As 21 twenty-firsts constitute one unit, 31 twenty-firsts constitute one unit and ten more twenty-firsts. 21)31 LEAST COMMON DENOMINATOR 29. The product of a number of factors is called a multiple of any one of them. Thus, 6 is a multiple of 3; 21 is a multiple of 7; and ab is a multiple of b. Now the denominators of any group of fractions must be reduced to some common multiple, called a common denominator, before the fractions may be added, subtracted, or compared in value. This common denominator may be obtained by multiplying together the denominators of the fractions. Suppose, for example, it is desired of a foot to fractional parts of a foot, having a 1 3 to reduce and 144 common denominator. Let the common denominator be 3 X 4 = 12. To reduce 193 to higher terms, i. e., to a fraction having the given denominator 12, multiply both numerator and denominator by the The least common denominator of several fractions is the least quantity which may be divided by each denominator without a remainder. Thus, 24 is the least common denominator of the fractions 5 3 6 8 7 and and ab is the least common denominator of the 1 12 1 fractions and b a The least common denominator of several fractions contains all of the prime factors of the given denominators. Thus, either 96 contains all their prime factors, 2, 2, 3, and 2, 2, 2, 2. The least common denominator, however, is neither 96 nor 192, but 48, because 48 is the least quantity which contains the prime factors of both 12 and 16 the greatest number of times that they appear in either one of these quantities. Since 2 is a common factor of 12 and 16 and appears four times in the latter, it is taken four times. There are no other common factors and therefore, the four 2's and the remaining factor 3, are all the factors which make up the least common denominator, and their product is 48. Therefore, to find the least common denominator, separate the denominators into their prime factors and take each factor as many times as it appears in the denominator containing it the greatest number of times. SOLUTION. First, write the denominators in a row, as shown in the margin. Now the least quantity to contain 3, 4, 9, and 12 must be the smallest quantity that will contain the factors of each of them, but no other factors. Then all the prime factors that the common denominator contains must be found. 2 is a prime factor of 4 and 12. Therefore, it must be a factor of any quantity that contains 4 and 12 without a remainder. Divide the 4 and 12 by 2, writing the quotients below, carrying down the 3 and 9 which are not divisible by 2. Again it is seen that 2 is a factor of 2 and 6, and the operation is repeated, obtaining 3, 1, 9, and 3. Next dividing by 3, the result is 1, 1, 3, and 1. these final quotients have no common factor, and must be factors of the least common denominator just as 2, 2, and 3 are. Disregarding the 1's, 2 × 2 × 3 × 3 = 36. The result 36 is the least common denominator. Whenever 1's appear, they may be disregarded, as multiplying by the factor 1 produces no change in the quantity. Now 2)3; 4; 9; 12 2)3; 2; 9; 6 3)3; 1; 9; 3 1; 1; 3; 1 of 6, and any quantity which has 12 as a factor must contain 6, it is only necessary to find the least common denominator of the denominators 12 and 7, which is 84. To reduce fractions to their least common denominator, first reduce each fraction to its lowest terms, and then find the least common denominator of these fractions. Each fraction must now be changed to a fraction whose denominator is the least common denominator, which operation, it is readily seen, is merely the reducing of the fractions to higher terms. For example, in the preceding paragraph, the least 4 common denominator of the fractions and is found to 7' 1 5 SOLUTION. 1 must be reduced to lower terms by dividing both numerator and denominator by 6, which changes the form to . The least common denominator of 1, 1, §, and √, has just been found to be 36. ΙΣ |