🇺🇸 AMC 8 ⇄ switch contest
1987 AJHSME

Problem 12

Problem 12 · 1987 AJHSME Hard
Geometry & Measurement fraction-of-rectangle
Figure for AJHSME 1987 Problem 12
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Answer: C — 1⁄12.
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Hint 1 of 2
Read the shaded block's two dimensions off the grid. You can compare it to the whole as a fraction without ever computing a big area.
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Hint 2 of 2
The shaded block spans 3 of the 18 units across and 6 of the 12 units up. What fraction of the width is that, and what fraction of the height?
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Approach: multiply the width-fraction by the height-fraction
  1. The black block is 3 units wide and 6 units tall. As a share of the whole rectangle that's 3⁄18 = 1⁄6 of the width and 6⁄12 = 1⁄2 of the height.
  2. A sub-rectangle's share of area is (its width-share) × (its height-share): 1⁄6 × 1⁄2 = 1⁄12.
  3. Why this transfers: shrinking each direction independently multiplies the areas — handling the two ratios separately keeps the numbers tiny and dodges 12 × 18 = 216 entirely.
Another way — areas directly:
  1. Shaded area = 3 × 6 = 18; whole = 12 × 18 = 216.
  2. 18 ⁄ 216 = 1⁄12.
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