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Some researchers needed to estimate how many catfish were in a local river, so they marked $280$ catfish with tags. Later, the researchers trapped $540$ catfish, of which $27$ had tags. To the nearest whole number, what is the best estimate for the catfish population?

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Estimate population size using proportions

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Q. Some researchers needed to estimate how many catfish were in a local river, so they marked

280

catfish with tags. Later, the researchers trapped

540

catfish, of which

27

had tags. To the nearest whole number, what is the best estimate for the catfish population?

Set up proportion: Step $1$ : Set up the proportion based on the capture-recapture method. $\newline$ Marked catfish found: $27$ $\newline$ Total catfish caught in second sample: $540$ $\newline$ Total catfish marked initially: $280$ $\newline$ Let $N$ be the estimated catfish population. $\newline$ The proportion is set as: $\newline$ $\frac{27}{540} = \frac{280}{N}$
Solve proportion: Step $2$ : Solve the proportion by cross-multiplying to find $N$ . $27 \times N = 280 \times 540$ Cross multiply: $27N = 151200$
Isolate N: Step $3$ : Divide both sides by $27$ to isolate $N$ . $\newline$ $N = \frac{151200}{27}$ $\newline$ $N = 5600$

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