We wanted to find what portion of the population owns a dog. First, we know that 60% of the population own at least one pet. Since all pets are randomly distributed, people with 1 pet must have a three fifths chance of owning a second pet, and so on. This makes the average number of pets per person equal to 3/5 + (3/5)2 + (3/5)3 + … . The general formula for a sequence like this is shown below. In this case, we get 1.5. Now, we make an equation. Let the ratio of dogs to cats to fish to people be x:y:z:1. We know that 3x = 2y, x+y+z = 1.5, and 6y / (1-y) = 5z / (1-z). We can easily substitute y to get x+1.5x+z = 1.5 and 9x / (1-1.5x) = 5z / (1-z). Substituting z is impossible. We must use a matrix. In this case it is long and difficult and we will not show it. The positive solution is x = 1/3, z = 2/3, and so y = 1/2. Since the ratio of dogs to people is 1:3, 1/(3+1) of the population must own dogs to make 1/(3+1-1) = 1/3. And so our answer is that 1/4 of the population owns at least 1 dog.
\sum_{k=1}^\infin\left(\frac ab \right)^k=\frac a{b-a}