AMC 8 · 2008 · #12

Grade 6 arithmetic

Archetype Branching State Enumeration └ Sub-archetype Numeric Operation Tree

sequences-geometricfraction-multiplicationpattern-recognition pattern-recognitionsystematic-enumeration ↑ Prerequisites: fraction-multiplicationpattern-recognition

📏 Short solution 💡 2 insights

Problem

A ball is dropped from 3 m. Its first bounce rises to 2 m. After that, each bounce rises to

\frac{2}{3}

of the previous bounce's height. Which numbered bounce is the first one that does not reach 0.5 m?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Spot a Pattern

Each bounce is a fixed $\frac{2}{3}$ of the one before, so the heights form a geometric pattern. Tool #5 (Spot a Pattern) names that rule and tells us we can keep multiplying by $\frac{2}{3}$ . Tool #2 (Make a List) is the cleanest way to use that rule: write out the bounce heights as fractions one by one and stop the first time the value drops below 0.5 m. With only five or six steps to check, a list is faster than setting up an inequality.

1STEP 1

Name the pattern — each bounce is $\frac{2}{3}$ of the one before, so from the first bounce the heights keep multiplying by $\frac{2}{3}$ .

h₁ = 2, h_n+1 =

\frac{2}{3}

h_n

2STEP 2

List the heights as exact fractions: 2, $\frac{4}{3}$ , $\frac{8}{9}$ , $\frac{16}{27}$ , then $\frac{32}{81}$ — kept exact so the compare to $\frac{1}{2}$ stays clean.

h₁ = 2, h₂ =

\frac{2}{3}

· 2 =

\frac{4}{3}

, h₃ =

\frac{2}{3}

\frac{4}{3}

\frac{8}{9}

, h₄ =

\frac{2}{3}

\frac{8}{9}

\frac{16}{27}

, h₅ =

\frac{2}{3}

\frac{16}{27}

\frac{32}{81}

3STEP 3

Compare each to $\frac{1}{2}$ via $\frac{a}{b}$ ＜ $\frac{1}{2}$ exactly when 2a ＜ b — only h₅ drops below, since 64 ＜ 81.

h₃: 2 · 8 = 16 ＞ 9, so

\frac{8}{9}

＞

\frac{1}{2}

. h₄: 2 · 16 = 32 ＞ 27, so

\frac{16}{27}

＞

\frac{1}{2}

. h₅: 2 · 32 = 64 vs. 81, and 64 ＜ 81, so

\frac{32}{81}

＜

\frac{1}{2}

4STEP 4

Bounces 1–4 all clear 0.5 m, but bounce 5 rises to only $\frac{32}{81}$ ≈ 0.395 m — so bounce 5 is the first to fall short.

First n with h_n ＜ 0.5 is n = 5 → (C)

Answer

Decimal check: h₁ = 2, h₂ ≈ 1.333, h₃ ≈ 0.889, h₄ ≈ 0.593, h₅ ≈ 0.395. The heights shrink by about 33% each bounce, so going from h₄ ≈ 0.593 to h₅ ≈ 0.395 crossing under 0.5 matches what we expect. The bounce numbering also matches: the 3-m drop is not counted, and bounce 1 is the 2-m rise, so bounce 5 is the fifth rebound, which is choice (C).

💡Key takeaway

When each step shrinks by the same fraction, listing the values is faster than algebra — multiply by $\frac{2}{3}$ , compare to $\frac{1}{2}$ , and stop at the first one that's too small.

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