Originally Posted by

**topsquark** $\displaystyle \frac{ \frac{1}{2} }{ \frac{3}{5} }$

We want to clear out the fractions in the numerator and denominator, so we need to multiply top and bottom by the LCM of 2 and 5, which is 10:

$\displaystyle = \frac{ \frac{1}{2} }{ \frac{3}{5} } \cdot \frac{2 \cdot 5}{2 \cdot 5}$

$\displaystyle = \frac{ \frac{1}{2} \cdot 2 \cdot 5 }{ \frac{3}{5} \cdot 2 \cdot 5 }$

$\displaystyle = \frac{1 \cdot 5}{3 \cdot 2} = \frac{1}{2} \times \frac{5}{3}$

This is, by far, the harder method to do the problem with, but it is the way to prove that the shortcut you learned is correct.

-Dan