Problem 16 · 1998 AJHSME
Medium
Ratios, Rates & Proportions
exponential-growthestimation
Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. Estimate the year in which the population will be about 6000.
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Answer: B — About 2075.
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Hint 1 of 2
Flip the previous question around: instead of 'how big after N steps,' ask 'how many ×3 steps until 200 grows to 6000?' First find how many times bigger 6000 is than 200.
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Hint 2 of 2
Build up the powers of 3 (3, 9, 27, 81, …) and see which one lands near your target multiple — then turn that count of steps into years.
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Approach: find the growth factor, match it to a power of 3, convert steps to years
- How many times the start is 6000? 6000 ÷ 200 = 30.
- Now count triplings to reach about 30×: ×3, ×9, ×27 — three triplings give 27, close to 30. (A fourth would overshoot to 81.) So three 25-year steps.
- Three steps = 3 × 25 = 75 years after 1998, landing on about 2075.
- Why this transfers: to find WHEN growth reaches a target, divide target by start to get the multiple, then count how many ×3's (the powers of 3) reach it — the same skill as reading a logarithm, just by hand.
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