Écologistes are trying to déterminé how many fish there are in a particular lake. They catch 100 fish from thé lake, put small tags on their tails, and put thème back into thé lake. Several dans later they catch 15 fish. They find that 2 are tagged ans the other 13 are not. Assuming that thé tagged to untagged fish are in the ratio 2:13, how many fish are in the lake (counting the 15)

Question

GPT-4o mini · Answer

To determine the total number of fish in the lake, we can use the information provided and the concept of proportions.

Let \( N \) be the total number of fish in the lake. When the ecologists tagged and released the fish, they tagged 100 fish. After several days, they caught 15 fish, 2 of which were tagged.

From the capture, we know the ratio of tagged to untagged fish is \( \frac{2}{13} \).

Thus, for every 2 tagged fish, there are 13 untagged fish, making a total of \( 2 + 13 = 15 \) fish.

We can set up the proportion based on the ratio obtained from the capture. If there are 2 tagged fish in the sample of 15:

\[ \frac{100}{N} = \frac{2}{15} \]

Now, we can cross-multiply to find \( N \):

\[ 100 \cdot 15 = 2 \cdot N \]

\[ 1500 = 2N \]

Now, divide both sides by 2:

\[ N = \frac{1500}{2} = 750 \]

Therefore, the estimated total number of fish in the lake is \( \boxed{750} \).