Problem 7 · 2011 AMC 8
Easy
Geometry & Measurement
fraction-of-area
Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?
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Answer: C — 25%.
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Hint 1 of 2
You don't need real measurements — just read each square as a fraction of itself: one of four equal columns is 1/4, a triangle filling half of a quarter is 1/8, and so on.
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Hint 2 of 2
Since all four squares are the same size, the answer is the average of the four shaded fractions: add them up and divide by 4.
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Approach: read each square as a fraction, then average the four
- Read the shaded part of each square: one of four columns = 1/4; a triangle that's half of one quarter = 1/8; a full quarter plus that triangle = 3/8; one full quarter = 1/4.
- These four fractions add to 1/4 + 1/8 + 3/8 + 1/4 = 2/8 + 1/8 + 3/8 + 2/8 = 8/8 = 1 — together they fill exactly one whole square.
- One whole square out of four equal squares is 1/4 = 25%.
- Intuition: the four shaded pieces, rearranged, would tile one complete square — that "they add to a whole" is what makes 25% pop out cleanly.
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