In a certain pond, 50 fish were caught, tagged, and returned to the pond. A few days later, 50 fish...

1. Translate the problem requirements: We need to find the total number of fish in the pond using the capture-recapture method. The key assumption is that the proportion of tagged fish in the second catch represents the same proportion of tagged fish in the entire pond.
2. Set up the proportional relationship: Establish that the ratio of tagged fish to total fish in the second catch equals the ratio of all tagged fish to the total pond population.
3. Apply the proportion to solve: Use cross-multiplication to find the total number of fish in the pond.
4. Verify reasonableness: Check that our answer makes sense given the sample data and matches one of the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what happened in plain English:

• First, 50 fish were caught, tagged (marked), and put back into the pond
• Later, another 50 fish were caught, and 2 of these had tags
• We're told that this sample represents the overall pond accurately

We need to find the total number of fish in the entire pond. The key insight is that if we caught a representative sample, then the percentage of tagged fish in our sample should match the percentage of tagged fish in the whole pond.

Process Skill: TRANSLATE - Converting the capture-recapture scenario into a proportion problem

2. Set up the proportional relationship

Here's the logical reasoning in everyday terms:

In our second catch:
• 2 out of 50 fish were tagged
• This means \(\frac{2}{50} = 4\%\) of the fish in our sample were tagged

If this sample represents the pond accurately, then:
• 4% of ALL fish in the pond should be tagged
• We know exactly how many tagged fish are in the pond: 50 (from the first catch)

So we can set up this relationship:
• If 50 tagged fish represents 4% of the total pond...
• Then the total pond = \(50 \div 0.04\)

In mathematical terms:
\(\frac{2}{50} = \frac{50}{\mathrm{Total\,fish\,in\,pond}}\)

3. Apply the proportion to solve

Using cross-multiplication on our proportion:

\(\frac{2}{50} = \frac{50}{\mathrm{Total}}\)

Cross-multiplying:
\(2 × \mathrm{Total} = 50 × 50\)
\(2 × \mathrm{Total} = 2,500\)
\(\mathrm{Total} = 2,500 ÷ 2\)
\(\mathrm{Total} = 1,250\)

Let's verify this makes sense:
• If there are 1,250 fish total, and 50 are tagged
• Then \(\frac{50}{1,250} = 4\%\) are tagged
• In our sample of 50 fish, 4% would be 2 fish ✓

4. Verify reasonableness

Let's check our answer against the given choices:
• Our calculated answer is 1,250 fish
• This matches choice (C) exactly

Reasonableness check:
• We tagged 50 fish and later found 2 tagged fish in a sample of 50
• This suggests tagged fish make up 4% of the population
• 50 tagged fish representing 4% of the total gives us 1,250 total fish
• This seems reasonable - not too small (which would make our sample unlucky) and not too large (which would make our sample too lucky)

Final Answer

The approximate number of fish in the pond is 1,250.

The correct answer is (C) 1,250.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what the proportion represents
Students often confuse which quantities to use in the proportion. They might set up the proportion as tagged fish in sample / total tagged fish = sample size / total population, which reverses the relationship. The correct setup should be: tagged fish in sample / sample size = total tagged fish / total population.

Faltering Point 2: Forgetting that all 50 originally tagged fish are still in the pond
Some students might think that only the 2 tagged fish found in the second catch represent the tagged fish in the pond, forgetting that all 50 fish from the first catch were returned to the pond and are still there. This leads them to use 2 instead of 50 as the total number of tagged fish.

Faltering Point 3: Misinterpreting the capture-recapture method
Students may not recognize this as a classic capture-recapture problem and instead try to solve it using other methods like simple addition or incorrect probability calculations, missing the key insight that the sample percentage should match the population percentage.

Errors while executing the approach

Faltering Point 1: Cross-multiplication errors
When cross-multiplying the proportion \(\frac{2}{50} = \frac{50}{\mathrm{Total}}\), students might incorrectly write it as \(2 × 50 = 50 × \mathrm{Total}\) instead of \(2 × \mathrm{Total} = 50 × 50\), leading to a final answer of 1,250/25 = 50 instead of 1,250.

Faltering Point 2: Arithmetic calculation mistakes
Students may correctly set up \(2 × \mathrm{Total} = 2,500\) but then make division errors, such as calculating \(2,500 ÷ 2 = 1,225\) or 1,275 instead of 1,250, especially when working under time pressure.

Errors while selecting the answer

Faltering Point 1: Choosing 2,500 instead of 1,250
Some students might correctly calculate \(2 × \mathrm{Total} = 2,500\) but then forget to complete the final division step, selecting answer choice (D) 2,500 instead of the correct answer (C) 1,250.