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Question 1
A lake has an area of 2 km². Scientists count 350 fish in this lake. What is the population density of fish in the lake?
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Correct answerCorrect!Incorrect
A175 fish/km²
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Recall the population density formula
Population density is calculated as: $Population Density = \frac{Number of individuals}{Area}$
Step 2: Substitute the values
Number of individuals $= 350$ fish, Area $= 2$ km². $Population Density = \frac{350}{2} = 175 fish/km^{2}$
Step 3: State the answer
The population density is 175 fish/km², meaning on average there are 175 fish for every square kilometre of the lake.
Method #2Process of Elimination
Step 1: Identify what is being calculated
Population density = number of individuals divided by area. We have 350 fish in 2 km².
Step 2: Eliminate 700 fish/km²
700 would result from multiplying $350 \times 2$ , which confuses multiplication with division. Density requires dividing by area, not multiplying.
Step 3: Eliminate 350 fish/km²
350 fish/km² would only be correct if the lake had an area of exactly 1 km². Since the lake is 2 km², this answer ignores the area entirely.
Step 4: Eliminate 1750 fish/km²
1750 would come from multiplying 350 by 5, which has no basis in this calculation. It vastly overestimates the density.
Step 5: Select 175 fish/km²
$\frac{350}{2} = 175$ fish/km². This correctly applies the density formula.
Question 2
A biologist marks 60 butterflies and releases them into a meadow. Two weeks later, she captures 90 butterflies, of which 18 are marked. Using the mark-recapture method, what is the estimated total butterfly population?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A300
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Recall the Lincoln-Petersen formula
The mark-recapture formula is: $N = \frac{M \times C}{R}$ where $M$ = number marked and released, $C$ = total caught in second sample, $R$ = recaptured marked individuals.
Step 2: Substitute the values
From the question: $M = 60$ , $C = 90$ , $R = 18$ . $N = \frac{60 \times 90}{18} = \frac{5400}{18} = 300$
Step 3: Confirm the answer
The estimated total butterfly population is 300 individuals. This means roughly 1 in 5 butterflies in the meadow was originally marked, consistent with the 18 out of 90 recaptured.
Method #2Process of Elimination
Step 1: Set up the calculation
We need $N = \frac{M \times C}{R} = \frac{60 \times 90}{18}$ . We can check each option by working backwards: if $N$ is correct, then $R = \frac{M \times C}{N}$ should equal 18.
Step 2: Eliminate 270
If $N = 270$ : $R = \frac{60 \times 90}{270} = \frac{5400}{270} = 20$ . This would mean 20 marked butterflies recaptured, but we are told only 18 were. Incorrect.
Step 3: Eliminate 150
If $N = 150$ : $R = \frac{5400}{150} = 36$ . This would require 36 out of 90 second-sample butterflies to be marked, far more than the 18 actually found. Incorrect.
Step 4: Eliminate 540
If $N = 540$ : $R = \frac{5400}{540} = 10$ . Only 10 marked individuals would be recaptured, but 18 were actually found. Incorrect.
Step 5: Select 300
$N = \frac{60 \times 90}{18} = 300$ . Checking: $R = \frac{5400}{300} = 18$ ✓. This matches the data exactly.

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Step-by-step walkthrough

Step 1: Recall the population density formula

Step 2: Substitute the values

Step 3: State the answer

Step 1: Identify what is being calculated

Step 2: Eliminate 700 fish/km²

Step 3: Eliminate 350 fish/km²

Step 4: Eliminate 1750 fish/km²

Step 5: Select 175 fish/km²

Step-by-step walkthrough

Step 1: Recall the Lincoln-Petersen formula

Step 2: Substitute the values

Step 3: Confirm the answer

Step 1: Set up the calculation

Step 2: Eliminate 270

Step 3: Eliminate 150

Step 4: Eliminate 540

Step 5: Select 300

12 more questions in this topic