Working out expression values What is the value of this expression?
$$\frac{1}{\dfrac{1}{\frac{1}{2} + \frac{1}{3}} + \dfrac{1}{\frac{1}{4} + \frac{1}{5}}}$$
I thought I'd start by working out 1/2 + 1/3,which is 5/6, and  then working out 1/4 + 1/5, which is 9/20. Then I added them together, which is 1 17/60. But how do I proceed?

As you've noticed, $\frac{1}{2}+\frac{1}{3}=\frac{5}{6}$ and $\frac{1}{4}+\frac{1}{5}=\frac{9}{20}$. Therefore:
$$\begin{align}
\cfrac{1}{
 \cfrac{1}{ \cfrac{1}{2}+\cfrac{1}{3} } + \cfrac{1}{ \cfrac{1}{4}+\cfrac{1}{5} }
} 
& =\cfrac{1}{ \cfrac{1}{ \cfrac{5}{6} }+\cfrac{1}{ \cfrac{9}{20} } }
\\[2ex] & =\cfrac{1}{\cfrac{6}{5}+\cfrac{20}{9}}
\\[2ex] & =\cfrac{1}{\cfrac{6\cdot 9+20\cdot 5}{5\cdot 9}}
\\[2ex] & =\cfrac{5\cdot 9}{6\cdot 9+20\cdot 5}
\\[2ex] & = \ldots
\end{align}$$
