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2016 AMC 10B Problem 10

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2016 AMC 10B Problem 10AMC 10 Step by Step
Name:                             Date: September 13, 2026
Similar & Congruent Triangles · difficulty 2 of 5 · about 2 min · position P1-10

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 33 inches weighs 1212 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of 55 inches. Which of the following is closest to the weight, in ounces, of the second piece?

  1. A)

    14.014.0

  2. B)

    16.016.0

  3. C)

    20.020.0

  4. D)

    33.333.3

  5. E)

    55.655.6

Answer

D

Key insight

Same wood and thickness means weight is proportional to area, and area of similar triangles scales by the square of the side ratio: 12 * (5/3)^2 = 100/3.

Solution

Both pieces have the same density and thickness, so their weights are proportional to their areas.

The two equilateral triangles are similar with side ratio 53\frac{5}{3}, so their areas are in the ratio (53)2=259\left(\frac{5}{3}\right)^2 = \frac{25}{9}.

Therefore the second piece weighs
12⋅259=3009=1003≈33.3 ounces.12 \cdot \frac{25}{9} = \frac{300}{9} = \frac{100}{3} \approx 33.3 \text{ ounces}.

The answer is (D) 33.3\boxed{\textbf{(D)}\ 33.3}.

Why this works

Weight of a uniform sheet is a stand-in for area, and area of similar figures grows with the square of the linear scale factor. You never need the actual area formula 34s2\frac{\sqrt3}{4}s^2; the ratio alone does the work. Look for the word "similar" (or identical shapes at different sizes) and immediately square the length ratio.

Alternative approach

Compute directly: area of the small triangle is 34⋅9\frac{\sqrt3}{4}\cdot 9, the large one 34⋅25\frac{\sqrt3}{4}\cdot 25; weight per unit area is 12/(934)12 / \left(\frac{9\sqrt3}{4}\right), and multiplying by the large area gives 12⋅259=33.3‾12 \cdot \frac{25}{9} = 33.\overline{3}.

The trap

Scaling the weight by the side ratio 5/3 (giving 20) instead of by its square 25/9.

Common mistakes

  • Scaling the weight by the side ratio 53\frac53 (giving 2020) instead of by its square 259\frac{25}{9}.
  • Cubing the ratio as if weight followed volume, giving 12⋅12527≈55.612 \cdot \frac{125}{27} \approx 55.6 (choice E); the thickness is the same, so only two dimensions scale.

Techniques

Set up the equation/formula and compute; no special trick needed