When two or more fractions do not have a common denominator, it is always possible to so change their forms as to make them have one. Thus, 3, 4, and can be changed (see § 92) into the forms, 37, and 18. As they now appear, they have the common denominator 36. 96. Lowest Common Denominator (L. C. D.). The two fractions and may be made to have a common denominator in many different ways. Thus we may write them as and, or as and, or as and 13, etc. Of all these ways the most important one (as we shall see) is that in which the common denominator is lowest. In other words, the most important way of writing and so that they shall have a common denominator is & and, respectively. Note that as thus written, the common denominator, 6, is simply the lowest (or least) common multiple (§ 87) of the two given denominators 2, 3. This one denominator, 6, is called the lowest common denominator of and . In general, two or more fractions may always be made to appear with their lowest common denominator in the way just mentioned. We have only to find the L. C. M. of the various denominators, and then change each fraction (by § 92) so that it will have this L. C. M. as its new denominator. For example, in dealing with the fractions the denominators are 3, 4, 9, and 12. The L. C. M. of these is 36. So the given fractions, when written with lowest common denominator, are What has just been said applies word for word to algebra, as is illustrated in the following examples: EXAMPLE 1. Change the fractions into equal fractions having their lowest common denominator. SOLUTION. The two denominators are 4 yz and y2z3 and the L. C. M. of these (as found by § 89) is 4 y2z3. Writing the first fraction with the denominator 4 y2z3 (by multiplying both its numerator and denominator by yz2) gives xyz2 4 y223 Similarly, writing the second fraction with the same denominator, 4 y2z3 (by multiplying both its numerator and denominator by 4) gives 8r into equal fractions having their lowest common denominator. SOLUTION. First factor the given denominators, thus writing the two fractions in the forms The L. C. M. of these denominators is therefore (by § 89) seen to be (a+3)(a−3)(a+6). To give the first fraction this denominator, we must multiply its numerator and denominator by (a+6). Similarly, to give the second fraction the new denominator we must multiply both its numerator and denominator by (a−3). The desired new forms are therefore The L. C. M. of the denominators is therefore (x-2)(x+2) (x+3)(x−3). Giving each of the fractions this denominator, they become (x−2)(x+2)(x+3)(x−3)' (x−2)(x+2)(x+3)(x−3)' 97. From these examples we have the following rule. RULE FOR REDUCING FRACTIONS TO THEIR LOWEST COMMON DENOMINATOR. Find the L. C. M. of the denominators. Then multiply both the numerator and denominator of each fraction by such an expression as will make that fraction have this L. C. M. as its denominator. ORAL EXERCISES First state the new denominator for each of the following fractions in order that they may have their lowest common denominator; then state the new forms for the fractions themselves. (Use the rule in § 97.) Write the fractions in each of the following exercises with lowest common denominator. (See Examples in § 96.) 98. Addition and Subtraction of Fractions. In adding and subtracting fractions in algebra we proceed as in arithmetic. Thus, to add and, we reduce them to 2 and 12, and the sum is then 1, or 1. Instead of reducing the fractions to the common denominator 12, we might reduce them to the common denominator 24; that is, to 12 and 11. In this case the sum is 34, or 11⁄2 as before. But it is better to do the first way; that is, to reduce to the lowest common denominator, and then add. a In the same way, in adding and we first reduce them b d' SOLUTION. Reducing the given fractions to their lowest common denominator, we have SOLUTION. Reducing the given fractions to their lowest common |
