AMC 10 · 2014 · #5
Grade 8 geometry-2dDoug 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?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A square window is built from 8 equal panes arranged in 2 rows and 4 columns. Each pane has height-to-width ratio $5:2$, and every border (around the outside and between panes) is $2$ inches wide. Find the side length of the square window.
Givens: There are $8$ equal panes, laid out as $2$ rows by $4$ columns; For each pane, height : width $= 5 : 2$; Every border, around and between the panes, is $2$ inches wide; The whole window is a square, so its width equals its height; Answer choices: (A) $26$, (B) $28$, (C) $30$, (D) $32$, (E) $34$
Unknowns: The side length of the square window, in inches
Understand
Restated: A square window is built from 8 equal panes arranged in 2 rows and 4 columns. Each pane has height-to-width ratio $5:2$, and every border (around the outside and between panes) is $2$ inches wide. Find the side length of the square window.
Givens: There are $8$ equal panes, laid out as $2$ rows by $4$ columns; For each pane, height : width $= 5 : 2$; Every border, around and between the panes, is $2$ inches wide; The whole window is a square, so its width equals its height; Answer choices: (A) $26$, (B) $28$, (C) $30$, (D) $32$, (E) $34$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #1 Draw a Diagram, #7 Identify Subproblems
The pane size is given only as a ratio $5:2$, so Tool #4 (Introduce a Variable) names one scale factor $x$ and turns the ratio into real lengths: width $2x$, height $5x$. Tool #1 (Draw a Diagram) reads the picture to count how many panes and borders lie along each side. Tool #7 (Identify Subproblems) splits the job into two totals — the width across and the height down — each built from panes plus borders. Setting those two totals equal (the window is square) gives one linear equation to solve for $x$.
Execute — Answer: A
6.RP.A.3 Step 1 Turn the ratio into lengths
- Each pane is taller than it is wide, in the ratio height : width $= 5 : 2$.
- Pick one scale factor $x$ and let the width be $2x$ and the height be $5x$.
- Then the ratio is $5x : 2x = 5 : 2$, exactly as required, and $x$ is the one unknown to find.
💡 A ratio only fixes the shape, not the size; one scale factor unlocks the actual measurements.
6.EE.A.2 Step 2 Add up the width across
- Look along the width: $4$ panes sit side by side, separated and framed by borders.
- With $4$ panes there are $5$ vertical borders — one on the far left, one on the far right, and $3$ between the panes.
- Each border is $2$ inches.
- So the width is $4$ pane-widths plus $5$ borders: $4(2x) + 5(2) = 8x + 10$.
💡 Lining up $n$ panes needs $n+1$ borders — one more border than there are panes.
6.EE.A.2 Step 3 Add up the height down
- Now look down the height: $2$ panes are stacked, framed by borders.
- With $2$ panes there are $3$ horizontal borders — top, middle, and bottom.
- So the height is $2$ pane-heights plus $3$ borders: $2(5x) + 3(2) = 10x + 6$.
💡 Stacking $2$ panes needs $3$ borders, again one more border than panes.
8.EE.C.7 Step 4 Use the square condition to solve
- The window is a square, so its width and height are equal.
- Set the two totals equal and solve: $8x + 10 = 10x + 6$.
- Subtract $8x$ from both sides to get $10 = 2x + 6$, then subtract $6$ to get $4 = 2x$, so $x = 2$.
💡 Being a square forces width and height to match, which is exactly the equation that pins down $x$.
6.EE.A.2 Step 5 Compute the side length
- Put $x = 2$ back into either total.
- The width is $8x + 10 = 8(2) + 10 = 26$, and as a check the height is $10x + 6 = 10(2) + 6 = 26$ — they agree, confirming the square.
- So the side length is $26$ inches, which is choice (A).
💡 Both totals landing on the same number is the built-in proof that the answer is right.
6.RP.A.3 Each pane is taller than it is wide, in the ratio height : width $= 5 : 2$. Pick 6.EE.A.2 Look along the width: $4$ panes sit side by side, separated and framed by border 6.EE.A.2 Now look down the height: $2$ panes are stacked, framed by borders. With $2$ pan 8.EE.C.7 The window is a square, so its width and height are equal. Set the two totals eq 6.EE.A.2 Put $x = 2$ back into either total. The width is $8x + 10 = 8(2) + 10 = 26$, and Review
Reasonableness: The two independent totals for width ($8x+10$) and height ($10x+6$) both evaluate to $26$ at $x=2$, which is exactly the square condition — a strong internal check. A sanity read of the picture agrees: with $x=2$ each pane is $4$ wide and $10$ tall, four wide panes plus five $2$-inch borders make $16+10=26$, and two tall panes plus three borders make $20+6=26$. The value $26$ is the smallest choice, matching a small scale factor $x=2$.
Alternative: Skip the algebra and test with concrete numbers using the equal-width-and-height idea. Try pane width $4$ and height $10$ (ratio $5:2$): width $= 4\cdot4 + 5\cdot2 = 26$ and height $= 2\cdot10 + 3\cdot2 = 26$. They match, so this is the square, and the side is $26$, confirming (A). If a guess made width and height disagree, you would rescale the pane until they matched.
CCSS standards used (min grade 8)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Turning the pane ratio $5:2$ into lengths $5x$ and $2x$ with a single scale factor.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Writing the width $8x+10$ and height $10x+6$ from counts of panes and borders, then evaluating them at $x=2$.)8.EE.C.7Solve linear equations in one variable (Solving $8x+10 = 10x+6$ (variable on both sides) to find $x=2$.)
⭐ Name the pane size with one variable, add panes and borders to get the width and height, then set them equal because the window is a square.
⭐ Name the pane size with one variable, add panes and borders to get the width and height, then set them equal because the window is a square.
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