2014 AMC 10B Problem 5

1. Read the picture. The $8$ panes sit in $4$ columns and $2$ rows.
2. Name the pane size. The height-to-width ratio is $5:2$, so say each pane is $2x$ inches wide and $5x$ inches tall.
3. Measure across. Going left to right you pass a border, a pane, a border, a pane, a border, a pane, a border, a pane, a border. That is $4$ panes and $5$ borders:
$$\text{width} = 4 \times 2x + 5 \times 2 = 8x + 10.$$
4. Measure down. Going top to bottom you pass a border, a pane, a border, a pane, a border. That is $2$ panes and $3$ borders:
$$\text{height} = 2 \times 5x + 3 \times 2 = 10x + 6.$$
5. Use "square." The width and the height are equal: $8x + 10 = 10x + 6$. Subtract $8x + 6$ from both sides to get $4 = 2x$, so $x = 2$.
6. The side is $8 \times 2 + 10 = 26$ inches. (Check with the height: $10 \times 2 + 6 = 26$.)

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

A ratio like $5:2$ means the two lengths are $5$ and $2$ times some unknown number, so one letter handles both. The word "square" gives the equation: two different ways of measuring the side must agree. The only careful part is counting borders; with $n$ panes in a row there are always $n + 1$ strips around and between them.

Try the choices. With side $26$, the glass across is $26 - 5 \times 2 = 16$ inches, so each pane is $16 \div 4 = 4$ inches wide. The glass down is $26 - 3 \times 2 = 20$ inches, so each pane is $20 \div 2 = 10$ inches tall. Since $10 : 4 = 5 : 2$, choice (A) works.

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 $\boxed{\textbf{(A)}\ 26}$.