Lily wanted to estimate the number of fish in a lake. She randomly captured 30 fish, then tagged and released them. One week later she randomly captured 40 fish from the same lake. She found that 12 of these 40 fish were tagged. What is the best estimate for the total number of fish in the lake?

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Lily wanted to estimate the number of fish in a lake - HSC - SSCE Mathematics Standard - Question 4 - 2022 - Paper 1

Question 4

Lily wanted to estimate the number of fish in a lake. She randomly captured 30 fish, then tagged and released them. One week later she randomly captured 40 fish fr... show full transcript

Worked Solution & Example Answer:Lily wanted to estimate the number of fish in a lake - HSC - SSCE Mathematics Standard - Question 4 - 2022 - Paper 1

Step 1

Calculate the proportion of tagged fish in the second sample

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Answer

To find the proportion of tagged fish in the second sample, divide the number of tagged fish found in this sample by the total number of fish captured in the second sample:

$\text{Proportion of tagged fish} = \frac{12}{40} = 0.3$

Step 2

Set up the proportion to estimate the total fish population

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Answer

Let $N$ be the total number of fish in the lake. The proportion of tagged fish in the lake (30 tagged fish) should equal the proportion of tagged fish in the second sample. Thus, we can set up the equation:

$\frac{30}{N} = 0.3$

Step 3

Solve for the total fish population

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Answer

Rearranging the equation to solve for $N$ gives:

$30 = 0.3N$

Dividing both sides by 0.3 leads to:

$N = \frac{30}{0.3} = 100$

Step 4

Conclusion

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Answer

The best estimate for the total number of fish in the lake is 100.

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