2500.11 – Pet Rocks

Our town has exactly 5000 families. Every family has 0, 1, or 2 pet rocks. Every pet rock belongs to a family. Most families have 1 pet rock. Exactly half of the remaining families have 2 pet rocks. How many pet rocks are there in Oberlin?

Solution

Solution by pure reasoning. Some families have 1 pet rock. Half of the rest have 2. The other half of the rest have none. If each family having 2 gave one of them to a family having none, then every family in Oberlin would have exactly 1 pet rock. So there are 5000 pet rocks in Oberlin.

Solution by algebra. If nn families have 1 pet rock, then (5000−n2)(\frac{5000 - n}{2}) have 2 pet rocks . So the number of pet rocks is

n+2(5000−n2)=n+5000−n=5000.n + 2(\frac{5000 - n}{2}) = n + 5000 - n = 5000.

A lovely problem.