2014 UNCO Math Contest II Problems/Problem 3

Problem

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


Solution

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~

See also

2014 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions