Competition · AMC preparation · step 4 of 4

AMC 8 · 2020 · #24

Grade 7 geometry-2dalgebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
area-rectanglespercentageratio-proportionlinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: area-rectanglespercentagelinear-equations-one-var
📏 Long solution 💡 4 insights 📊 Diagram
Problem
A big square is tiled with n × n gray squares, each of side s inches, separated and bordered by strips of width d inches. The picture in the problem shows the layout for n=3. For n=24 the 576 gray tiles together cover 64% of the big square's area. Find the ratio ds\frac{d}{s}.

Pick an answer.

(A)
$\frac{6}{25}$
(B)
$\frac{1}{4}$
(C)
$\frac{9}{25}$
(D)
$\frac{7}{16}$
(E)
$\frac{9}{16}$

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

n = 24 is too big to count borders by hand, so start with Tool #9 and use the n = 3 picture the problem already gives. Counting tiles vs. borders in that small case exposes the pattern "n tiles, n+1 borders" along one side; Tool #1 (the diagram is already supplied) makes the n+1 borders obvious — one between each pair, plus one at each end. Once the side length is ns + (n+1)d, Tool #13 turns the percent condition into a single linear equation in d and s that we can solve for the ratio.

1STEP 1

Count pieces along one side

In the n=3 picture one side has 3 tiles and 4 border strips, so its length is 3s + 4d.

side(n=3) = 3s + 4d
2STEP 2

Write the general side count

For general n a side has n tiles and n+1 borders, so its length is ns + (n+1)d; at n=24 this becomes 24s + 25d.

side(n) = n s + (n+1) d → side(24) = 24s + 25d
3STEP 3

Write both square areas

The 576 tiles form a 24s-wide gray square of area (24s)², sitting in a big square of side 24s + 25d and area (24s + 25d)².

A_gray = (24s)², A_big = (24s + 25d)²
4STEP 4

Turn 64 percent into a ratio

For similar squares, area ratio = (side ratio)²; since 64% = (45\frac{4}{5})², the gray side is 45\frac{4}{5} of the big side: 24s24s+25d\frac{24s}{24s + 25d} = 45\frac{4}{5}.

A_gray/A_big = 64/100 = (4/5)² → 24s/(24s + 25d) = 4/5
5STEP 5

Cross-multiply and solve

Cross-multiply: 120s = 96s + 100d, so 24s = 100d and ds\frac{d}{s} = 625\frac{6}{25}, choice (A).

5 · 24s = 4(24s + 25d) → 120s = 96s + 100d → 24s = 100d → d/s = 24/100 = 6/25 → (A)
Answer
6/25
Sanity-check the side ratio directly: the gray block is 24s wide and the big square is 24s + 25d wide, so the gray fraction (per side) should be 45\frac{4}{5} = 0.8. With ds\frac{d}{s} = 625\frac{6}{25}, one side is 24s + 25 · 625\frac{6}{25}s = 24s + 6s = 30s, and 24s30s\frac{24s}{30s} = 45\frac{4}{5}. Squaring gives 1625\frac{16}{25} = 0.64 = 64% — exactly the given coverage, so 625\frac{6}{25} is consistent.
💡Key takeaway

This AMC 8 problem only needs Grade 7 scale-drawing reasoning — areas of similar squares scale by the square of the side ratio — that you already know!

  • Count pieces along one side
  • Write the general side count
  • Write both square areas
  • Turn 64 percent into a ratio
  • Cross-multiply and solve

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