Which of the following is the correct order of the fractions $$\dfrac{15}{11}$$, $$\dfrac{19}{15}$$ and $$\dfrac{17}{13}$$, from least to greatest? ($$2019$$ AMC $$8$$ Problem, Question \#$$3$$) 
Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering
Method $$1$$:  Each one is $$x+\dfrac{4}{x}$$ so we are really comparing $$\dfrac{4}{11}$$, $$\dfrac{4}{15}$$, and $$\dfrac{4}{13}$$, where you can see $$\dfrac{4}{11}$$ $$\textgreater$$ $$\dfrac{4}{13}$$ $$\textgreater$$ $$\dfrac{4}{15}$$. So, $$\dfrac{19}{15}\textless{} \dfrac{17}{13}\textless{} \dfrac{15}{11}$$.  Method $$2$$:  We take a common denominator:  $$\dfrac{15}{11}, \dfrac{19}{15}, \dfrac{17}{13}= \dfrac{15 \cdot 15 \cdot 13}{11 \cdot 15 \cdot 13}, \dfrac{19 \cdot 11 \cdot 13}{15 \cdot 11 \cdot 13}, \dfrac{17 \cdot 11 \cdot 15}{13 \cdot 11 \cdot 15}= \dfrac{2925}{2145}, \dfrac{2717}{2145}, \dfrac{2805}{2145}$$.  $$2717 \textless{} 2805\textless{} 2925$$. So, $$\dfrac{19}{15}\textless{} \dfrac{17}{13}\textless{} \dfrac{15}{11}$$.  Method $$3$$:  When $$\frac{x}{y}\textgreater1$$, $$\frac{x+z}{y+z}\textless\frac{x}{y}$$.  Hence, $$\dfrac{19}{15}\textless{} \dfrac{17}{13}\textless{} \dfrac{15}{11}$$. 