To model the problem mathematically, we aim to create a linear equation that represents the total cost of purchasing the buses while respecting the given budget. The equation can be derived as follows:

1. Let $x$ represent the number of type I buses (costing Rs. 300,000 each).
2. Let $y$ represent the number of type II buses (costing Rs. 500,000 each).
3. The total cost of purchasing these buses is given by: $300,000x + 500,000y$
4. The budget allocated for this purchase is Rs. 50 million, which we write as 50,000,000.

Required function:

The required function representing the total cost in terms of $x$ and $y$ is: $300,000x + 500,000y \leq 50,000,000$

Alternatively, this can be simplified by dividing through by 1,000 to make the equation more manageable: $300x + 500y \leq 50,000$

This is the budget constraint function for the problem.

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If additional information is needed (e.g., maximizing the number of buses, constraints on $x$ and $y$, etc.), further equations or inequalities can be added.

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Would you like to:

1. Explore a graphical representation of the function?
2. Solve for $x$ and $y$ under specific conditions?
3. Formulate additional constraints for $x$ and $y$?
4. Maximize the number of buses within the budget?
5. Analyze feasible solutions for integers only?

Tip: Breaking large numbers into smaller scales (e.g., dividing by 1,000) simplifies calculations and visualization.