AMC 10 · 2016 · #10

Grade 7 geometry-2d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series
similar-figuresratio-proportionarea-triangles dimensional-analysis ↑ Prerequisites: similar-figuresratio-proportion
📏 Medium solution 💡 2 insights
Problem
A thin equilateral triangle of wood with side length 3 inches weighs 12 ounces. Another piece of the same wood, same thickness, is an equilateral triangle with side length 5 inches. Find the weight, in ounces, closest to that of the second piece.

Pick an answer.

(A)
14.0
(B)
16.0
(C)
20.0
(D)
33.3
(E)
55.6

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Because the wood, thickness, and density are identical, the weight of each piece rides entirely on its flat area — tool #8 (Analyze the Units) tells us "more area means proportionally more ounces." Both pieces are equilateral triangles, so they are similar, and a flat shape scaled in length by a factor k has its area scaled by k². That turns the whole problem into a single proportion: name the unknown weight w with tool #4 (Introduce a Variable) and set w12\frac{w}{12} equal to the area ratio. Tool #3 (Eliminate Possibilities) then matches the computed weight to the closest choice.

1STEP 1

Weight tracks area

Same wood, density, and thickness — only the flat area changes, so weight is proportional to area; just compare the two areas.

weight ∝ area (same wood, same thickness)
2STEP 2

Area scales by the square of the side

Both are equilateral, hence similar; scaling the side 3 to 5 stretches every length by 53\frac{5}{3}, so area grows by (53\frac{5}{3})² = 259\frac{25}{9}.

area2area1\frac{area₂}{area₁} = √(3)4⋅52√(3)4⋅32\frac{\frac{√(3)}{4}· 5²}{\frac{√(3)}{4}· 3²} = 259\frac{25}{9}
3STEP 3

Set up the proportion

Let w be the second triangle's weight in ounces; since weight tracks area, the weight ratio equals the area ratio: w12\frac{w}{12} = 259\frac{25}{9}.

w12\frac{w}{12} = 259\frac{25}{9}
4STEP 4

Solve and pick the closest choice

Multiply by 12: w = 12 · 259\frac{25}{9} = 3009\frac{300}{9} ≈ 33.3 ounces — nowhere near 14.0, 16.0, 20.0, or 55.6, so the answer is (D).

w = 12 · 259\frac{25}{9} = 3009\frac{300}{9} ≈ 33.3 = (D)
Answer
33.3
The side grew from 3 to 5, less than doubling, so the area factor 259\frac{25}{9} ≈ 2.78 is under 3 but well above 2. Multiplying 12 ounces by that factor should land between 24 and 36, and 33.3 sits right there. The tempting wrong answer 20.0 comes from scaling weight by the side ratio 53\frac{5}{3} instead of its square, and 55.6 comes from scaling by (53\frac{5}{3})³ as if the thickness also grew — both are excluded by the constant thickness.
💡Key takeaway

When a flat shape keeps the same thickness, its weight follows its area — and scaling every length by k multiplies the area, and the weight, by k².

  • Weight tracks area
  • Area scales by the square of the side
  • Set up the proportion
  • Solve and pick the closest choice

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