Difference between revisions of "2016 AMC 10B Problems/Problem 7"

Problem

The ratio of the measures of two acute angles is $5:4$, 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?

$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$

Solution

We can set up a system of equations where $x$ and $y$ are the two acute angles. WLOG, assume that $x$ $<$ $y$ in order for the complement of $x$ to be greater than the complement of $y$. Therefore, $5x$ $=$ $4y$ and $90$ $-$ $x$ $=$ $2$ $(90$ $-$ $y)$. Solving for $y$ in the first equation and substituting into the second equation yields $$\begin{split} 90 - x & = 2 (90 - 1.25x) \\ 1.5x & = 90 \\ x & = 60 \end{split}$$

Substituting this $x$ value back into the first equation yields $y$ $=$ $75$, leaving $x$ $+$ $y$ equal to $\boxed{\textbf{(C)}\ 135}$.

(Solution by akaashp11)

Solution 2

We let the measures be $5x$ and $4x$ giving us the ratio of $5:4$. We know $90-4x>90-5x$ since this inequality gives $x>0$, which is true since the measures of angles are never negative. We also know the bigger complement is twice the smaller, so

$90-4x=2(90-5x)$

$90-4x=180-10x$

$6x=90$

$x=15$

Therefore, the angles are $75$ and $60$, which sum to $\boxed{\textbf{(C)}\ 135}$.

~savannahsolver