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Some situations require us to compare fractions, that is, to rank them in order of size. For instance, suppose that Or to take another example, suppose that the drinking water in your home, according to a lab report, has 2 parts per million (ppm) of lead. Is the water safe to drink? If the federal limit on lead in drinking water is 15 parts per billion (ppb), again you need to compare fractions. One way to handle such problems is to draw diagrams corresponding to the fractions in question. The larger fraction corresponds to the larger shaded region. For instance, the diagrams show that Both Note that the symbols < and > always point to the smaller number. For like fractions, the fraction with the larger numerator is the larger fraction. Definitions Like fractions are fractions with the same denominator. Unlike fractions are fractions with different denominators. To Compare Fractions
EXAMPLE 1 Compare Solution These fractions are unlike because they have different denominators. Therefore we need to express them as equivalent fractions having the same denominator. But what should that denominator be? One common denominator that we can use is the product of the denominators: 15 · 9 = 135
Next, we compare the numerators of the like fractions that we just found. Because Another common denominator that we can use is the least common multiple of the denominators. The LCM is Because Note that in Example 1 we computed the LCM of the two denominators. This type of computation is used frequently in working with fractions. Definition For any set of fractions, their least common denominator (LCD) is the least common multiple of their denominators. In Example 2, pay particular attention to how we use the LCD. EXAMPLE 2 Order from smallest to largest: Solution Because these fractions are unlike, we need to find equivalent fractions with a common denominator. Lets use their LCD as that denominator. The LCD We write each fraction with a denominator of 40. Then we order the fractions from smallest to largest. (The symbol < stands for less than.) EXAMPLE 3 About Solution We need to compare Since |