[tex]4\dfrac{1}{2}-3\dfrac{5}{6}\\\\\text{Find the common denominator.}\\\\6=2\cdot3.\ \text{Therefore the common denominator is 6}.\\\\\dfrac{1}{2}=\dfrac{1\cdot3}{2\cdot3}=\dfrac{3}{6}\\\\\Downarrow\\\\4\dfrac{1}{2}-3\dfrac{5}{6}=4\dfrac{3}{6}-3\dfrac{5}{6}=(*)\\\\Method\ \#1:\\\\(*)=(4-3)+\left(\dfrac{3}{6}-\dfrac{5}{6}\right)=1+\left(-\dfrac{2}{6}\right)=\dfrac{6}{6}-\dfrac{2}{6}=\dfrac{4}{6}=\dfrac{4:2}{6:2}=\boxed{\frac{2}{3}}\\\\Method\ \#2\\\\(*)=3\dfrac{6+3}{6}-3\dfrac{5}{6}=3\dfrac{9}{6}-3\dfrac{5}{6}=\dfrac{4}{6}=\boxed{\dfrac{2}{3}}[/tex]
