2014 UNCO Math Contest II Problems/Problem 3

Problem

Find $x$ and $y$ if $\frac{1}{1+\frac{1}{x}}=2$ and $\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2$

Solve for each variable.

$\frac{1}{1+\frac{1}{x}}=2$ $1 = 2 + \frac{2}{x}$ $-1 = \frac{2}{x}$ $\boxed{x=-2}$

Now for $y$ :

$\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2$

$1 = 2 + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}$

$-1 = + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}$

$-1 - \frac{1}{1+\frac{1}{y}} = 2$

$\frac{1}{1+\frac{1}{y}} = -3$

$-4 = \frac{3}{y}$

$\boxed{y = -\frac{3}{4}}$

~IYN~

2014 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All UNCO Math Contest Problems and Solutions