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To calculate the gross income in a month, we first need to establish any details about Anne's monthly income or revenue streams provided in the problem. However, from the given information, no specific monthly income is mentioned directly, so the monthly rental income will be our primary source.

### Given Details:
1. Anne sells a house after a year for $13,000.
2. She rents an apartment for $500 per month.

If there are other income sources or a salary that Anne earns, it is not mentioned in the problem. Thus, if only rental costs and income from selling the house are considered, we cannot directly calculate gross income without more information.

**Please clarify or provide more details if Anne has an annual salary, investments, or other incomes**

This problem involves calculating gross income with a mix of hire purchase, compound interest, and budgeting. It also covers installment payments, profit from property sale, and monthly rent, providing a comprehensive look into personal finance. Suitable for Grades 10-12.

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