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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 $$ \frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \frac{1}{x}}} } } \ \ = \ \ \frac{67}{96} \ \ , $$

  • You can try working your way up 'unwrapping' the fraction into a single numerator and denominator. – Fimpellizzeri May 13 '16 at 02:50
  • If you unwrap the continued fraction on the left as Fimpellizieri suggests you should obtain an equation of the form $\dfrac{ax+b}{cx+d}=\dfrac{67}{96}$ which will be easily solved. – John Wayland Bales May 13 '16 at 02:53
  • Thank you very much! The thinking process helps a great deal! –  May 13 '16 at 02:57

2 Answers2

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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.

MPW
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    +1 By far the better way to do this problem, since we never have more than one occurrence of $x$ at any step of the working. –  May 13 '16 at 03:22
  • @lastresort: Agreed, thanks. When you can see that a certain sequence of invertible changes has been applied to the variable to get a specific value, it's always quite easy to just reverse the process to see where you came from. – MPW May 13 '16 at 03:26
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Start from the bottom. $$\frac 1{4+\frac 1x}=\frac 1{\frac{4x+1}x}=\frac x{4x+1}$$and more of the same

Ross Millikan
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