What is the correct ordering of the three numbers $$\dfrac{5}{19}$$, $$\dfrac{7}{21}$$, and $$\dfrac{9}{23}$$, in increasing order? ($$2012$$ AMC $$8$$ Problem, Question \#$$4$$) 
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Method $$1$$:  The value of $$\dfrac{7}{21}$$ is $$\dfrac{1}{3}$$. Now we give all the fractions a common denominator.  $$\dfrac{5}{19} \Rightarrow \dfrac{345}{1311}$$,  $$\dfrac{1}{3} \Rightarrow \dfrac{437}{1311}$$,  $$\dfrac{9}{23} \Rightarrow \dfrac{513}{1311}$$.  Ordering the fractions from least to greatest, we find that they are in the order  listed. Therefore, $$\frac{5}{19}\textless{} \frac{7}{21}\textless{} \frac{9}{23}$$.  Method $$2$$:  Instead of finding the LCD, we can subtract each fraction from $$1$$ to get a common numerator. Thus,  $$1- \dfrac{5}{19}= \dfrac{14}{19}$$,  $$1- \dfrac{7}{21}= \dfrac{14}{21}$$,  $$1- \dfrac{9}{23}= \dfrac{14}{23}$$.  All three fraction have the common numerator $$14$$. Now the order of the fractions is obvious. $$\dfrac{14}{19}\textgreater\dfrac{14}{21}\textgreater\dfrac{14}{23}\Rightarrow\dfrac{5}{19}\textless\dfrac{7}{21}\textless\dfrac{9}{23}$$. Therefore, $$\dfrac{5}{19}\textless\dfrac{7}{21}\textless\dfrac{9}{23}$$. 