Problem 17 · 1998 AJHSME
Hard
Ratios, Rates & Proportions
exponential-growthestimation
Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. The total area is 24,900 square miles, and the Queen requires at least 1.5 square miles per person. In about how many years from 1998 will the population reach the maximum the islands can support?
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Answer: C — About 100 years.
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Hint 1 of 2
Two stages here: first decide the target (the most people the land can hold), then count triplings to reach it. The cap comes from sharing the land: total area ÷ space-per-person.
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Hint 2 of 2
Once you have the cap, it's the same engine as before — divide by 200 to get the multiple, then count the ×3 steps (powers of 3) and turn them into years.
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Approach: compute the land's cap, then count triplings to reach it
- Largest population the land allows = total area ÷ space each person needs = 24,900 ÷ 1.5 = 16,600 people.
- That's 16,600 ÷ 200 = 83 times today's count. Walk up the powers of 3: 3, 9, 27, 81 — four triplings give 81, just shy of 83.
- Four steps = 4 × 25 = 100 years after 1998.
- Why this transfers: many word problems are a 'find the goal, then count steps to it' combo. Here the goal hides behind a sharing/division (area ÷ per-person), and the step-counting is the same powers-of-3 idea from the earlier parts.
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