Problem 3 - Biology Concepts and Applications

Question

Suppose 200 fish are marked and released in a pond. The following week, 200 fish are caught and 100 of them have marks. How many fish are in this pond?

Step-by-Step Solution

Answer

The pond has approximately 400 fish.

Step 1: Understanding the Problem

We need to determine the total number of fish in the pond. We know that 200 fish are marked and released. Later, 200 fish are caught, and of those, 100 are marked. This suggests a way to estimate the total number of fish using a ratio or proportion.

Step 2: Setting Up the Proportion

We can set up a proportion based on the assumption that marked and unmarked fish are evenly distributed in the population. Let \[ N \] be the total number of fish in the pond. The proportion of marked fish in the total catch is equal to the proportion of marked fish in the entire pond. So, we set up the equation: \[ \frac{100}{200} = \frac{200}{N} \]

Step 3: Solving the Proportion for Total Fish

To find \( N \), cross-multiply in the proportion: \[ 100 \times N = 200 \times 200 \]This simplifies to:\[ 100N = 40000 \]

Step 4: Finding the Value of N

Divide both sides of the equation by 100 to solve for \( N \): \[ \frac{100N}{100} = \frac{40000}{100} \]So, \[ N = 400 \].

Step 5: Conclusion

The estimated total number of fish in the pond is 400.

Key Concepts

Population Estimation

Population estimation is a statistical method used to infer the number of individuals in a broad population based on a sample. In wildlife and environmental studies, researchers often use the **capturing-recapture method** to estimate animal populations. This involves capturing a sample of organisms, marking them, and releasing them back into the population. After some time, another sample is captured, and the number of marked individuals within this new sample is counted.

For example, in the problem given, 200 fish were initially marked and released. A week later, another group of 200 fish was caught, of which 100 had marks. These two observations are vital in estimating the total fish population in the pond. By assuming that the proportion of marked fish caught in the second sample reflects their proportion in the total pond, we can calculate the total number using a straightforward proportion.

The concept of population estimation allows ecologists and scientists to make informed decisions about managing wildlife resources, even without the ability to count each individual in a habitat. This quick estimation method relies on the randomness and mixing of the animal population between the marked and recaptured events.

Proportion

A proportion is a statement that two ratios or fractions are equal. It provides a powerful way to solve for unknown quantities if you know the other terms in the relationship.

In the context of our fish problem, the proportion is established between the fraction of marked fish in the second sample of 200 and the entire pond population. When we assume the sample accurately represents the population, we can equate these ratios to solve for our unknown, which is the total number of fish in the pond. Here, the proportion is:

• Marked fish in sample = Marked fish in entire pond, represented as \( \frac{100}{200} = \frac{200}{N} \)

Recognizing proportions in these scenarios allows us to apply our knowledge of ratios to solve real-world problems efficiently, offering practicality in numerous scientific and mathematical challenges.

Cross-multiplication

Cross-multiplication is a simple mathematical process that allows for the solution of equations set up as proportions. When you have an equation that equates two fractions or ratios, you can "cross-multiply" to find an unknown quantity. This means you multiply the numerator of each fraction by the denominator of the other, essentially eliminating the fractions and creating an equation that is often easier to solve.

For the fish problem, after setting up the proportion \(\frac{100}{200} = \frac{200}{N} \), cross-multiplication is employed. The equation becomes:

• \( 100 \times N = 200 \times 200 \)

To find \(N\), simplify by dividing both sides by 100, resulting in:

• \( N = \frac{40000}{100} = 400 \)

Utilizing cross-multiplication is an effective strategy in solving proportions, and it’s particularly useful in fields where estimation is frequently necessary, such as ecology, stock assessment, and even in everyday decision-making scenarios.