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Lowest common denominator
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== Role in arithmetic and algebra == The same fraction can be expressed in many different forms. As long as the ratio between numerator and denominator is the same, the fractions represent the same number. For example: : <math>\frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}</math> because they are all multiplied by 1 written as a fraction: : <math>\frac{2}{3}=\frac{2}{3}\times\frac{3}{3}=\frac{2}{3}\times\frac{6}{6}=\frac{2}{3}\times\frac{72}{72}=\frac{2}{3}\times\frac{100,000}{100,000}.</math> It is usually easiest to add, subtract, or compare fractions when each is expressed with the same denominator, called a "common denominator". For example, the numerators of fractions with common denominators can simply be added, such that <math>\frac{5}{12}+\frac{6}{12}=\frac{11}{12}</math> and that <math>\frac{5}{12}<\frac{11}{12}</math>, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what <math>\frac{5}{12}+\frac{11}{18}</math> equals, or whether <math>\frac{5}{12}</math> is greater than or less than <math>\frac{11}{18}</math>. Any common denominator will do, but usually the lowest common denominator is desirable because it makes the rest of the calculation as simple as possible.<ref name=worldbook/>
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