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A farmer wanted to estimate how many moles were living on his property, so he marked $39$ of them with tags. A few weeks later, the farmer drove around his property and noticed $21$ tagged moles out of the $420$ moles he counted. To the nearest whole number, what is the best estimate for the mole population?

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Estimate population size using proportions

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Q. A farmer wanted to estimate how many moles were living on his property, so he marked

39

of them with tags. A few weeks later, the farmer drove around his property and noticed

21

tagged moles out of the

420

moles he counted. To the nearest whole number, what is the best estimate for the mole population?

Set up proportion: Step $1$ : Set up the proportion based on the capture-recapture method. $\newline$ We know: $\newline$ - Tagged moles found: $21$ $\newline$ - Total moles counted in second capture: $420$ $\newline$ - Moles originally tagged: $39$ $\newline$ Let $M$ be the estimated total mole population. $\newline$ Using the proportion: $\newline$ $(\text{tagged moles found} / \text{total moles counted}) = (\text{moles originally tagged} / \text{estimated total mole population})$ $\newline$ $\frac{21}{420} = \frac{39}{M}$
Solve proportion: Step $2$ : Solve the proportion by cross-multiplying to find $M$ . $\newline$ Cross-multiplying gives: $\newline$ $21 \times M = 39 \times 420$
Continue solving: Step $3$ : Continue solving for $M$ . $21 \times M = 16380$ $M = \frac{16380}{21}$ $M = 780$

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