SECTION XX. ADDITION AND SUBTRACTION OF FRACTIONS. COMMON DENOMINATOR. ART. 67. The addition and subtraction of fractions are represented by writing them after each other with the sign+and α between them, thus, —+ m care y n being taken to place the signs even with the line which separates the numerator from the denominator. But when the denominators are alike, we may perform the addition or subtraction upon the numerators, placing the result over the common denominator. 2 Thus, + α and Suppose it were required to add In this m case the denominators are different; but if the numerator and denominator of the first fraction be multiplied by n, and the numerator and denominator of the second fraction be multiplied by m, the denominators of the fractions will be made alike, while the value of the fractions remain un 1 y Let it be required to add, and. First, we reduce the fractions to a common denominator. If the numerator and denominator of each fraction be multiplied by the denominators of both the others, which does not change the value of the fractions, they become amy bey b my b my the sum of which is amy+bcy+bmx Ans. Hence we derive a RULE FOR THE ADDITION AND SUBTRACTION OF FRACTIONS. Reduce them to a common denominator, then add the numerators, or subtract one from the other, placing the result over the common denominator. ART. 68. From the preceding examples, we derive also the following RULE FOR REDUCING FRACTIONS TO A COMMON DENOMINATOR. Multiply all the denominators together for a common lenominator, and multiply each numerator by all the denominators except its own, in order to obtain the numer ators. The results obtained by this rule are correct, but not always the simplest. Let us reduce x 3 y 3a2, 4 ab' and 5 m to a common de6 63 c nominator. By taking the product of all the denominators, we shall obtain a common denominator considerably greater than is necessary. In this case, the least common denominator will be, as in arithmetic, the least common multiple of the given denominators. The least common multiple of 3 a2, 4 ab, and 663 c is 3.22 a2b3c, cr 12 a2b3c, which is the least common denominator sought. To produce 12 a2 63 c, the first denominator is multiplied by 4 63 c, the second by 3 a b2c, and the third by 2 a2; these are, therefore, the quantities by which the respective numerators are to be multiplied, and are evidently obtained by dividing the common denominator by each of the given denominators separately. This multiplication being performed, and the results placed over the common denomi463 cx 9 a b2 cy and 12 a2 b3 c' 12 a2 b3 c' nator, the fractions become ART. 69. Hence we have the following 10 a2 m 12 a2 b3 c RULE ΤΟ REDUCE FRACTIONS TO THE LEAST COMMON DENOMINATOR. Find the least common multiple of all the given denominators, and this will be the least common denominator; then divide the common denominator by each of the given denominators, and multiply the numerators by the respective quotients, placing each of these products over the common denominator. Remark. Fractions must be simplified before applying 18. Reduce 22, that is, 2+, to the form of a fraction. Since 4 fourths make 1, 2=; and &+1=4, Ans. In like manner, 7—3—5—1=4. Reducing b to a fraction having d for a denominator, ART. 70. How many times is contained in 9? Since in 9 there are 6, is contained 63 times in 9, and is contained as many times, that is, words, 9 divided by gives $3 for a quotient. is the same as the product of 9 by 7. How many times is contained in a? 3; in other The result Since in a number a of units there are 5 a fifths, is contained 5 a times in a, and is contained as many |