Scholars caught, marked and released 80 fishes in a pond. Later 100 fish were caught at random. 40 of them were marked. The number of fishes in the pond is -

Options

A. 400
B. 200
C. 100
D. 50

Answer

To solve the problem of estimating the total number of fishes in the pond, we can use the method of capture-recapture. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Variables
Let \( x \) be the total number of fishes in the pond.

### Step 2: Set Up the Ratios
From the information given:
- 80 fishes were caught, marked, and released.
- Later, 100 fishes were caught at random, out of which 40 were marked.

We can set up the ratios based on the marked fishes:
- The ratio of marked fishes to total fishes when the first group was marked is:
\[ \frac{80}{x} \]
- The ratio of marked fishes to total fishes when the second group was caught is:
\[ \frac{40}{100} \]

### Step 3: Equate the Ratios
Assuming that the marked fishes are evenly distributed in the pond, we can equate the two ratios:
\[ \frac{80}{x} = \frac{40}{100} \]

### Step 4: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 80 \times 100 = 40 \times x \]
This simplifies to:
\[ 8000 = 40x \]

### Step 5: Solve for \( x \)
Now, divide both sides by 40:
\[ x = \frac{8000}{40} = 200 \]

### Conclusion
The total number of fishes in the pond is \( 200 \).

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