AMC 8 · 2001 · #17
Grade 6 rate-ratiopatternProblem
For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the following table (where K = 1000).
\begin{tabular}{rccccccccccccccc} \text{Question} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \text{Value} & 100 & 200 & 300 & 500 & 1\text{K} & 2\text{K} & 4\text{K} & 8\text{K} & 16\text{K} & 32\text{K} & 64\text{K} & 125\text{K} & 250\text{K} & 500\text{K} & 1000\text{K} \end{tabular}
Between which two questions is the percent increase of the value the smallest?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Prize values: $100, 200, 300, 500, 1\text{K}, 2\text{K}, 4\text{K}, 8\text{K}, 16\text{K}, 32\text{K}, 64\text{K}, 125\text{K}, 250\text{K}, 500\text{K}, 1000\text{K}$ for questions $1$ through $15$; $1\text{K} = 1000$; Answer choices: (A) From $1$ to $2$, (B) From $2$ to $3$, (C) From $3$ to $4$, (D) From $11$ to $12$, (E) From $14$ to $15$
Unknowns: Which of the five listed question-pairs has the smallest percent increase in prize value
Understand
Restated: 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.
Givens: Prize values: $100, 200, 300, 500, 1\text{K}, 2\text{K}, 4\text{K}, 8\text{K}, 16\text{K}, 32\text{K}, 64\text{K}, 125\text{K}, 250\text{K}, 500\text{K}, 1000\text{K}$ for questions $1$ through $15$; $1\text{K} = 1000$; Answer choices: (A) From $1$ to $2$, (B) From $2$ to $3$, (C) From $3$ to $4$, (D) From $11$ to $12$, (E) From $14$ to $15$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check
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.
Execute — Answer: B
6.RP.A.3 Step 1 - Write down the percent-increase formula in the form we will reuse for every pair.
- The base of the percent is always the older (smaller-numbered) question.
💡 Treating the older value as the "whole" (the base of $100\%$) is the Grade 6 percent-change convention.
6.RP.A.3 Step 2 - Apply the formula to the five listed pairs.
- Read the prize values straight off the table and reduce each fraction.
💡 For (D), the values are in thousands but the units cancel in the ratio, so $\tfrac{125-64}{64}$ uses the same arithmetic as cents or dollars.
6.NS.C.7 Step 3 - Line up the five percents and pick the smallest.
- The list is $100\%,\;50\%,\;66.7\%,\;95.3\%,\;100\%$, and $50\%$ is clearly the minimum.
💡 Ordering five rational numbers and picking the smallest is a Grade 6 number-line comparison.
6.RP.A.3 Write down the percent-increase formula in the form we will reuse for every pair 6.RP.A.3 Apply the formula to the five listed pairs. Read the prize values straight off t 6.NS.C.7 Line up the five percents and pick the smallest. The list is $100\%,\;50\%,\;66. Review
Reasonableness: 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).
Alternative: Tool #5 (Look for a Pattern): most of the prize table doubles each question ($100 \to 200 \to 400$ would be perfect doubling), so the percent increase is usually about $100\%$. The only place the table breaks the doubling pattern is the soft step $200 \to 300$ (only $+50\%$) and the slightly soft step $64\text{K} \to 125\text{K}$ (about $+95\%$). Of those two off-pattern jumps, the bigger break — and therefore the smaller percent increase — is from question $2$ to $3$, giving (B).
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, including percent problems (Applying the percent-increase formula $\tfrac{V_{\text{new}}-V_{\text{old}}}{V_{\text{old}}}\times 100\%$ to each of the five listed pairs.)6.NS.C.7Understand ordering and absolute value of rational numbers (Comparing the five computed percents ($100\%, 50\%, 66.7\%, 95.3\%, 100\%$) to pick the smallest.)
⭐ When a prize jumps by the same dollars, the percent change shrinks as the starting value grows — so $\$200 \to \$300$ is a $50\%$ jump, the smallest of the five pairs.
⭐ When a prize jumps by the same dollars, the percent change shrinks as the starting value grows — so $\$200 \to \$300$ is a $50\%$ jump, the smallest of the five pairs.