AMC 8 · 2001 · #17

Grade 6 rate-ratiopattern

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

percentagefraction-arithmeticratio-proportionsequences-geometric systematic-enumerationidentify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

On Who Wants To Be A Millionaire?, the prize for each of 15 questions doubles or steps up by an irregular amount. Among five listed pairs of consecutive questions, find the pair whose prize jump is the smallest percent increase.

Pick an answer.

(A)

From 1 to 2

(B)

From 2 to 3

(C)

From 3 to 4

(D)

From 11 to 12

(E)

From 14 to 15

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question has five separate candidates, so Tool #7 (Identify Subproblems) splits the work cleanly: compute one percent increase per candidate, then compare. Tool #6 (Guess and Check) is the right second tool because we are literally testing each choice against the formula and keeping the smallest. No algebra is needed — just the percent-increase formula five times and a head-to-head comparison.

1STEP 1

Set up the percent-increase formula, always dividing by the older (smaller-numbered) question's prize.

Percent increase = (V_new - V_old)/V_old × 100%

2STEP 2

Plug each pair into the formula and reduce to get 100%, 50%, about 66.7%, about 95.3%, and 100%.

(A)

\frac{200-100}{100}

= 1 = 100% (B)

\frac{300-200}{200}

\frac{1}{2}

= 50% (C)

\frac{500-300}{300}

\frac{2}{3}

≈ 66.7% (D)

\frac{125-64}{64}

\frac{61}{64}

≈ 95.3% (E)

\frac{1000-500}{500}

= 1 = 100%

3STEP 3

Line up the five percents and pick the smallest: 50% wins.

min(100%, 50%, 66.7%, 95.3%, 100%) = 50% → (B)

Answer

From 2 to 3

The added dollar amount is not what decides the winner — the base matters. (A) adds only

100 but on a base of

100, so the percent is huge (100%). (B) also adds

100, but the base is

200, cutting the percent in half to 50%. Every other listed pair at least doubles (100%) or comes close to doubling (95.3%), so 50% stands out as the only "sub-doubling" jump. This matches answer (B).

💡Key takeaway

When a prize jumps by the same dollars, the percent change shrinks as the starting value grows — so $200 →$ 300 is a 50% jump, the smallest of the five pairs.

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