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2014 AMC 10B Problem 5

2014 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

Camden 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?

A)
$26$
B)
$28$
C)
$30$
D)
$32$
E)
$34$

Answer

Key insight

With panes 2x by 5x in a 4-by-2 grid, width 8x + 10 must equal height 10x + 6, forcing x = 2.

Solution

The figure shows the $8$ panes in $4$ columns and $2$ rows. Let each pane be $2x$ wide and $5x$ tall.

Across the window there are $4$ pane widths and $5$ border strips (one on each edge and three between columns):
$\text{width} = 4(2x) + 5(2) = 8x + 10.$

Down the window there are $2$ pane heights and $3$ border strips:
$\text{height} = 2(5x) + 3(2) = 10x + 6.$

The window is square, so $8x + 10 = 10x + 6$ , giving $x = 2$ . The side length is $8(2) + 10 = 26$ .

The answer is $\boxed{\textbf{(A)}\ 26}$ .

Why this works

A ratio like $5:2$ is a scale factor: write the dimensions as $5x$ and $2x$ and let the geometry supply the equation. Here "square" means two independent expressions for the side must agree. The only care needed is counting border strips: $n$ panes in a row always sit among $n + 1$ strips.

Alternative approach

Test choice (A): side $26$ leaves $26 - 10 = 16$ inches of glass across, so panes are $4$ wide, and $26 - 6 = 20$ inches of glass down, so panes are $10$ tall. The ratio $10 : 4 = 5 : 2$ checks.

The trap

Miscounting the borders: the width has five 2-inch strips (two outer, three between) and the height has three.

Common mistakes

Miscounting the borders: the width has five 2-inch strips (two outer, three between) and the height has three.
Reading the ratio backwards (panes $5$ wide and $2$ tall), which yields no consistent square.

Techniques

Set up the equation/formula and compute; no special trick needed