How to solve an optimization problem that deals with the optimal combination of given ratios?
If the number of children $n$ is fixed and there is no limit on the number of staff members you can potentially hire, here's one way you can maximize the amount the government pays you:

You can use linear programming (LP).  Let $s_1$ denote the number of staff you hire to manage one-child classrooms; $s_2$ the number of staff you hire to manage two-child classrooms; and $s_3$ the number of staff you hire to manage three-child classrooms.  The number of children you can handle, with this configuration of staff, is $s_1 + 2s_2 + 3s_3$.  Also, the amount paid by the government is $7.28 s_1 + 9.10 s_2 + 14.71 s_3$.  You want to maximize the payoff, subject to the constraints $s_1 \ge 0$, $s_2 \ge 0$, $s_3 \ge 0$ and to the constraint $s_1 + 2 s_2 + 3 s_3 \ge n$ (i.e., you have capacity to take care of $n$ children).

Of course, this formulation is silly, because under this formulation the way to maximize your profit is to maximize the number of staff chosen.  That's because this formulation doesn't take into account the cost of hiring staff or any other consideration.  So you'll need to adjust it to incorporate the additional constraints you have.

Fortunately, this LP formulation makes it easy to incorporate additional constraints.  For instance, suppose there are only 100 qualified people who you'd possible be willing to hire as staff, so you simply can't hire more than a total of 100 staff members.  That can be handled by adding the additional constraint $s_1 + s_2 + s_3 \le 100$.

In practice there's probably some cost per staff member you hire, so in practice we probably want to maximize not the amount the government pays us, but the amount of "profit" left over after paying the staff members.  Suppose each staff member costs 5 dollars per hour.  Then you would adjust the above by maximizing the revenue minus the costs, i.e., $(7.28 s_1 + 9.10 s_2 + 14.71 s_3) - (5.00 s_1 + 5.00 s_2 + 5.00 s_3)$.

In practice you want to only hire an integer number of staff, so you might want to use an integer linear programming (ILP) solver rather than a LP solver.

I still question whether this is the right metric for a non-profit to be maximizing.  Instead of focusing only on "profit", I would anticipate that a non-profit would care about the benefit to stakeholders or soceity: e.g., the benefit to the children served, or the number of children served, or some combination of benefit and leftover "profit".  For instance, all else being equal, is the benefit greater if a child is cared by a single staff member rather than sharing a staff member with two other children? I suspect so, but that isn't incorporated in your "profit maximization" formulation.  Thus, I suspect the right objective function to maximize might be something different, chosen to take into account that the goal of a non-profit is not to derive a profit but to accrue some social benefit.