Problem 17 · 2001 AMC 8
Hard
Fractions, Decimals & Percents
percent-multiplierestimate-and-pick
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). Between which two questions is the percent increase of the value the smallest?
| Question | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Value | 100 | 200 | 300 | 500 | 1K | 2K | 4K | 8K |
| Question | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|
| Value | 16K | 32K | 64K | 125K | 250K | 500K | 1000K |
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Answer: B — From 2 to 3.
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Hint 1 of 2
Don't compute every percent — scan the pattern. A value that doubles is exactly a +100% increase, and most steps in this table double.
Still stuck? Show hint 2 →
Hint 2 of 2
So you can throw out every doubling and only compare the few steps that grow by less than double: 2→3, 3→4, and 11→12.
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Approach: ignore the doublings, compare the exceptions
- Recognize the pattern: doubling = +100%. Almost every step here doubles, so it can't be the smallest. Only three steps break the pattern: 2→3, 3→4, and 11→12 (and 11→12 is nearly a double, so it's large).
- Percent increase = (rise ÷ original). 2→3: 100/200 = +50%; 3→4: 200/300 ≈ +67%; 11→12: 61K/64K ≈ +95%.
- Smallest is from question 2 to 3 at 50%. The transferable insight: percent change compares the rise to the STARTING value, so spotting which steps grow proportionally least beats grinding every number.
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