2014 AMC 10B Problem 5

Below is the professionally curated solution for Problem 5 of the 2014 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 10B solutions, or check the answer key.

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Concepts:ratio and proportion linear equation

Difficulty rating: 1280

Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2 ,$ and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? \t\t

$\ 26$

$\ 28$

$\ 30$

$\ 32$

$\ 34$

Solution:

Let the smaller side of a pane be a distance of $x.$

Then, the side length is $5\cdot 2 + 4x.$ Also, the other direction has pane lengths of $2.5x.$

This means the side length is $3\cdot 2 + 2(2.5x) .$ We solve for $x$ as follows: $\begin{align*}10+4x &= 6+5x \\ x&= 4\end{align*}$

Therefore, the side length is $10+4\cdot 4 = 26.$

Thus, the correct answer is A .

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