Final answer:
The steady-state rate of single students in the dormitory is such that the transitions into and out of being single and in a relationship are equal. When the proportions stabilize, 50% of the students are single and 50% are in a relationship at steady-state.
Step-by-step explanation:
To calculate the steady-state rate of single students in a college dormitory we need to set up equations representing the transitions of students between being in a relationship and being single. Let's call S the proportion of single students and R the proportion of students in a relationship. We know that every month, 10% of students in a relationship break up and 5% of single students establish a new relationship.
The steady-state occurs when the rate of change in each group is zero. This means that the transitions into and out of each group are equal. In mathematical terms, we can set up the equilibrium equations as follows:
- For students in a relationship: 0.05S = 0.10R
- For single students: 0.10R = 0.05S
Since S + R = 1 (or 100% of the students), we can solve these equations simultaneously to find the steady-state proportions. They are in fact the same equation and thus we already know that in steady-state, S = R. Therefore,
S = R = 0.5 or 50% of the students are single at steady-state.