AMC 8 · 2010 · #13

Grade 6 geometry-2d

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

perimetersequences-arithmeticpercentagelinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: perimeterpercentagelinear-equations-one-var

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Problem

A triangle has three side lengths (in inches) that are three consecutive integers. The shortest side is exactly 30% of the perimeter. Find the length of the longest side.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

The reference solution uses Tool #13 (Algebra): set up n = 0.3(3n+3) and solve. But the answer choices give us the longest side directly, and each choice pins down all three sides. So Tool #6 (Guess and Check) is faster and needs no variable: for each choice L ∈ {7,8,9,10,11}, the three sides are L-2, L-1, L, the perimeter is 3L - 3, and we just check whether shortest = 0.3 × perimeter. Tool #2 (Systematic List) keeps the five cases organized; Tool #3 (Eliminate Possibilities) crosses off any choice that misses the 30% target. This path uses only multiplication and basic decimals — no algebra.

1STEP 1

If the longest side is L, the three sides are L-2, L-1, L and the perimeter is 3L-3; test whether L-2 = 0.3 × (3L-3).

sides = (L-2, L-1, L), P = 3L - 3, test: L - 2 ?= 0.3 (3L - 3)

2STEP 2

Try (A) L=7: sides (5,6,7), perimeter 18, but 0.3 × 18 = 5.4 ≠ 5, so reject (A).

0.3 × 18 = 5.4 ≠ 5 (A) rejected

3STEP 3

Try L=8, 9, 10: perimeters 21, 24, 27 give 0.3× = 6.3, 7.2, 8.1, none matching the shortest side, so reject (B), (C), (D).

L=8:& 0.3× 21 = 6.3 ≠ 6 ; L=9:& 0.3× 24 = 7.2 ≠ 7 ; L=10:& 0.3× 27 = 8.1 ≠ 8

4STEP 4

Try (E) L=11: sides (9,10,11), perimeter 30, and 0.3 × 30 = 9 exactly matches the shortest side, so the longest side is 11.

L = 11: sides (9,10,11), P=30, 0.3× 30 = 9 = shortest ✓ → (E)

Answer

If the three sides were equal, each side would be

\frac{1}{3}

≈ 33.3% of the perimeter. Here the shortest side is only 30% — a bit less than a third — so the sides must be a little spread apart, not tightly bunched. The winning triple (9, 10, 11) has a shortest side 9 out of a perimeter 30, and

\frac{9}{30}

= 0.30 exactly. Triangle inequality also holds: 9 + 10 = 19 ＞ 11. Everything checks out.

💡Key takeaway

This AMC 8 problem only needs Grade 6 percent reasoning — "30% of the perimeter" — that you already know!

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