Question

A biologist wants to calculate the number of fish in a lake. On May $1$ she catches a random sample of $60$ fish, tags them, and releases them. On September $1$ she catches a random sample of $70$ fish and finds that $3$ of them are tagged. To calculate the number of fish in the lake on May $1,$ she assumes that $25\%$ of these fish are no longer in the lake on September $1$ (because of death and emigrations), that $40\%$ of the fish were not in the lake May $1$ (because of births and immigrations), and that the number of untagged fish and tagged fish in the September $1$ sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May $1?$

Answer

Answer: 840

Concepts:percentage ratio and proportion sampling without replacement

Difficulty rating: 1830

Small Hint:

First determine how many of the original tagged fish remain in September

Big Hint:

Use the sample’s tagged fraction to estimate the September population, then identify the $60\%$ that were present in May

Solution:

Of the $60$ tagged fish, $75\%$ remain in September, so $45$ tagged fish remain. The sample estimates that tagged fish form $\frac{3}{70}$ of the September population, making that population $45\left(\frac{70}{3}\right)=1050.$ Of those fish, $60\%$ were present in May, so $630$ surviving May fish remain. These are $75\%$ of the May population. Thus the estimated May population is $\frac{630}{0.75}=840.$

1990 AIME Problem 6

Answer: 840

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years