AMC 10 · 2014 · #5

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
ratio-proportionlinear-equations-one-var convert-to-algebra ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
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 the panes, is 2 inches wide. Find the side length of the square window, in inches.

Pick an answer.

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

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

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.

1STEP 1

Turn the ratio into lengths

A ratio fixes shape, not size, so name one scale factor x: each pane is 2x wide and 5x tall.

pane width = 2x, pane height = 5x
2STEP 2

Add up the width across

Across the width sit 4 panes and 5 borders, one more border than panes: width = 4(2x) + 5(2) = 8x + 10.

width = 4(2x) + 5(2) = 8x + 10
3STEP 3

Add up the height down

Down the height sit 2 panes and 3 borders — top, middle, bottom: height = 2(5x) + 3(2) = 10x + 6.

height = 2(5x) + 3(2) = 10x + 6
4STEP 4

Use the square condition to solve

Square means width = height, so 8x + 10 = 10x + 6, giving 2x = 4 and x = 2.

8x + 10 = 10x + 6 → 4 = 2x → x = 2
5STEP 5

Compute the side length

With x = 2 both totals agree: 8(2) + 10 = 26 = 10(2) + 6, so the side is 26 inches, choice (A).

side = 8(2) + 10 = 26 = 10(2) + 6 → (A)
Answer
26
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.
💡Key takeaway

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.

  • Turn the ratio into lengths
  • Add up the width across
  • Add up the height down
  • Use the square condition to solve
  • Compute the side length

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