Problem 13 · 2010 AMC 8
Medium
Algebra & Patterns
consecutive-integerspercent-equation
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 30% of the perimeter. What is the length of the longest side?
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Answer: E — 11 inches.
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Hint 1 of 2
Consecutive integers are evenly spaced, so the middle side is exactly the average — meaning the perimeter is 3 times the middle side. That single fact replaces a lot of algebra.
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Hint 2 of 2
When three numbers are evenly spaced, their sum is 3 × the middle one. Now ‘shortest = 30% of perimeter’ becomes a clean comparison.
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Approach: use the evenly-spaced middle term
- Sides are m−1, m, m+1 around a middle value m. Their sum (the perimeter) is exactly 3m.
- The shortest side is 30% of the perimeter: m−1 = 0.30 · 3m = 0.9m. So 0.1m = 1, giving m = 10.
- Longest = m+1 = 11.
- Why this transfers: for any evenly-spaced list, swapping in ‘sum = (count) × middle’ collapses messy sums to one variable. It's the same idea behind averaging a run of consecutive numbers.
Another way — test the answer choices:
- Longest options are 7…11; try the longest = 11, so sides are 9, 10, 11 with perimeter 30.
- Is the shortest 30% of 30? 0.30 · 30 = 9 = shortest. It fits, so the longest side is 11 — a fast check when you'd rather verify than solve.
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