Problem 5 · 2023 AMC 8
Stretch
Ratios, Rates & Proportions
proportionratio
A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
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Answer: B — 1500 fish.
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Hint 1 of 2
Think of the net of 180 fish as a shrunk-down copy of the whole lake — same recipe, smaller pot. So the trout fraction matches.
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Hint 2 of 2
Find the trout fraction in the sample (30 out of 180), then apply that same fraction to the 250 known trout to recover the whole.
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Approach: the trout fraction is the same in sample and lake
- The sample is a tiny scale model of the lake: the trout fraction in your net should match the trout fraction in the whole lake. So find that one fraction and apply it.
- In the sample, 30 of 180 are trout: 30 ÷ 180 = 16. That clean fraction is the heart of the problem — trout are 1 in every 6 fish.
- So the 250 real trout are 16 of the whole lake, meaning the lake holds 250 × 6 = 1500 fish. This transfers to every ‘capture sample’ (or poll, or survey): part-of-sample = part-of-whole.
Another way — scale the whole sample up:
- The lake has 250 trout but the sample only caught 30 — so the lake is 250 ÷ 30 = 253 times as ‘trout-rich’ as the sample.
- Everything scales by that same factor, so total fish = 180 × 253 = 60 × 25 = 1500.
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