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Terrence is a scientist who works with the Department of Agriculture. In order to ensure a good harvest, Terrence keeps track of various local caterpillar populations. He catches $750$ caterpillars, marks them, and releases them. Then later, he catches $400$ caterpillars, $12$ of which are marked. To the nearest whole number, what is the best estimate for the caterpillar population?

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Estimate population size using proportions

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Q. Terrence is a scientist who works with the Department of Agriculture. In order to ensure a good harvest, Terrence keeps track of various local caterpillar populations. He catches

750

caterpillars, marks them, and releases them. Then later, he catches

400

caterpillars,

12

of which are marked. To the nearest whole number, what is the best estimate for the caterpillar population?

Set up proportion: Set up the proportion based on the mark and recapture data. $\newline$ Marked caterpillars found: $12$ $\newline$ Total caterpillars in second catch: $400$ $\newline$ Total caterpillars marked initially: $750$ $\newline$ Let $p$ be the estimated caterpillar population. $\newline$ The proportion is: $\newline$ $\frac{\text{marked caterpillars found}}{\text{total caterpillars in second catch}} = \frac{\text{total caterpillars marked initially}}{\text{estimated caterpillar population}}$ $\newline$ $\frac{12}{400} = \frac{750}{p}$
Solve for p: Solve for $p$ by cross-multiplying. $\newline$ $12 \cdot p = 750 \cdot 400$ $\newline$ $12p = 300000$
Divide to isolate p: Divide both sides by $12$ to isolate $p$ . $\newline$ $p = \frac{300000}{12}$ $\newline$ $p = 25000$

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