Difference between revisions of "2016 AMC 10B Problems/Problem 7"
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− | + | We can set up a system of equations where <math>x</math> and <math>y</math> are the two acute angles. WLOG, assume that <math>x</math> <math><</math> <math>y</math> in order for the complement of <math>x</math> to be greater than the complement of <math>y</math>. Therefore, <math>5x</math> <math>=</math> <math>4y</math> and <math>90</math> <math>-</math> <math>x</math> <math>=</math> <math>2</math> <math>(90</math> <math>-</math> <math>y)</math>. Solving for <math>y</math> in the first equation and substituting into the second equation yields | |
− | <math>5x</math> <math>=</math> <math>4y</math> | + | <cmath>\begin{split} |
+ | 90 - x & = 2 (90 - 1.25x) \\ | ||
+ | 1.5x & = 90 \\ | ||
+ | x & = 60 | ||
+ | \end{split}</cmath> | ||
+ | Substituting this <math>x</math> value back into the first equation yields <math>y</math> <math>=</math> <math>75</math>, leaving <math>x</math> <math>+</math> <math>y</math> equal to <math>\textbf{(C)}\ 135</math>. |
Revision as of 12:02, 21 February 2016
Problem
The ratio of the measures of two acute angles is , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
Solution
We can set up a system of equations where and are the two acute angles. WLOG, assume that in order for the complement of to be greater than the complement of . Therefore, and . Solving for in the first equation and substituting into the second equation yields Substituting this value back into the first equation yields , leaving equal to .