find a value of x in a fraction

Question

Can anyone help me with this? I don't know where to start. I assume there is a trick.

Fine the value of x if $$ \frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \frac{1}{x}}} } } \ \ = \ \ \frac{67}{96} \ \ , $$

Answer 1

Start from the bottom. $$\frac 1{4+\frac 1x}=\frac 1{\frac{4x+1}x}=\frac x{4x+1}$$and more of the same

Answer 2

Start with the given expression on the left and $67/96$ on the right, then invert, subtract $1$, invert, subtract $2$, invert, subtract $3$, invert, subtract $4$, and invert (on both sides).

The left side becomes $x$, and the right side becomes $2$. The result is $x=2$.

The explicit steps are, on the right side, the numbers

67/96,

96/67, 29/67,

67/29, 9/29,

29/9, 2/9,

9/2, 1/2,

2.