Difference between revisions of "2014 UNCO Math Contest II Problems/Problem 3"
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== Solution == | == Solution == | ||
| + | |||
| + | Solve for each variable. | ||
| + | |||
| + | <cmath>\frac{1}{1+\frac{1}{x}}=2</cmath> <cmath>1 = 2 + \frac{2}{x}</cmath> <cmath>-1 = \frac{2}{x}</cmath> <cmath>\boxed{x=-2}</cmath> | ||
| + | |||
| + | Now for <math>y</math>: | ||
| + | |||
| + | <cmath>\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2</cmath> | ||
| + | |||
| + | <cmath>1 = 2 + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}</cmath> | ||
| + | |||
| + | <cmath>-1 = + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}</cmath> | ||
| + | |||
| + | <cmath>-1 - \frac{1}{1+\frac{1}{y}} = 2</cmath> | ||
| + | |||
| + | <cmath>\frac{1}{1+\frac{1}{y}} = -3</cmath> | ||
| + | |||
| + | <cmath>-4 = \frac{3}{y}</cmath> | ||
| + | |||
| + | <cmath>\boxed{y = -\frac{3}{4}}</cmath> | ||
| + | |||
| + | ~IYN~ | ||
== See also == | == See also == | ||
| − | {{ | + | {{UNCO Math Contest box|year=2014|n=II|num-b=2|num-a=4}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
and 
if 
and 
![\[\frac{1}{1+\frac{1}{x}}=2\]](http://latex.artofproblemsolving.com/f/c/5/fc5a521412d762cd12269197619236638cdd13d4.png)
![\[1 = 2 + \frac{2}{x}\]](http://latex.artofproblemsolving.com/b/d/9/bd930824d4842bc690b7a6ab8056bd6303b6c545.png)
![\[-1 = \frac{2}{x}\]](http://latex.artofproblemsolving.com/1/c/3/1c39bf7831201df75ea4512932893f8f4a431726.png)
![\[\boxed{x=-2}\]](http://latex.artofproblemsolving.com/4/8/8/488283de1634a8345d583625af8bb2f1d7c630f1.png)
![\[\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2\]](http://latex.artofproblemsolving.com/9/c/9/9c96ff02aa743804eab57787491152ef84309385.png)
![\[1 = 2 + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}\]](http://latex.artofproblemsolving.com/f/6/9/f697aa67c8475db9c52bf187af536786e4aa2a7f.png)
![\[-1 = + \frac{2}{1+\frac{1}{1+\frac{1}{y}}}\]](http://latex.artofproblemsolving.com/6/0/5/6056df50954f5ef0fb8def912e85c7a96a1394ba.png)
![\[-1 - \frac{1}{1+\frac{1}{y}} = 2\]](http://latex.artofproblemsolving.com/f/c/5/fc55502e2289b0eff7679a48f270d8a1d6a83b45.png)
![\[\frac{1}{1+\frac{1}{y}} = -3\]](http://latex.artofproblemsolving.com/0/f/4/0f4e12f3dedd3f251b65689db82751e64cb6efde.png)
![\[-4 = \frac{3}{y}\]](http://latex.artofproblemsolving.com/a/9/c/a9ced3e1d8e3fd1853d09494e565c33dfd782d7f.png)
![\[\boxed{y = -\frac{3}{4}}\]](http://latex.artofproblemsolving.com/7/5/b/75b428189a54740843e20577bc244d771395cb34.png)
