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To solve this problem, let's break it down step-by-step:

### Problem Statement
- **Total Apartments**: 96
- **Current Rent per Apartment**: $550 per month
- **Current Occupancy**: 96 apartments are fully occupied.
- **Income Goal**: $54,600 per month.
- **Vacancy Condition**: For every $25 increase in rent, 3 apartments will become vacant.

### Goal
We need to find the rent per apartment that will result in the total monthly income of $54,600.

### Step-by-Step Solution

1. **Define Variables**:
   - Let the number of $25 increments be $ x $.
   - New rent per apartment: $ 550 + 25x $.
   - Number of rented apartments after $ x $ increments: $ 96 - 3x $.

2. **Total Income Formula**:
   Total Income = (Number of rented apartments) × (New rent per apartment)

   So, the total income is:
   \[
   (96 - 3x) \times (550 + 25x)
   \]

3. **Set up the Equation**:
   To meet the monthly income goal of $54,600, set up the equation:
   \[
   (96 - 3x)(550 + 25x) = 54,600
   \]

4. **Expand the Equation**:
   Expanding the left side:
   \[
   96 \times 550 + 96 \times 25x - 3x \times 550 - 3x \times 25x = 54,600
   \]
   \[
   52,800 + 2,400x - 1,650x - 75x^2 = 54,600
   \]
   Simplify:
   \[
   52,800 + 750x - 75x^2 = 54,600
   \]
   \[
   -75x^2 + 750x + 52,800 = 54,600
   \]
   \[
   -75x^2 + 750x - 1,800 = 0
   \]
   Divide through by -75:
   \[
   x^2 - 10x + 24 = 0
   \]

5. **Solve the Quadratic Equation**:
   \[
   x^2 - 10x + 24 = 0
   \]
   Using the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = -10 $, and $ c = 24 $:
   \[
   x = \frac{10 \pm \sqrt{(-10)^2 - 4 \times 1 \times 24}}{2 \times 1}
   \]
   \[
   x = \frac{10 \pm \sqrt{100 - 96}}{2}
   \]
   \[
   x = \frac{10 \pm 2}{2}
   \]
   \[
   x = \frac{12}{2} = 6 \quad \text{or} \quad x = \frac{8}{2} = 4
   \]

6. **Calculate Rent for Each Scenario**:
   - If $ x = 4 $: Rent = $ 550 + 25 \times 4 = 650 $.
   - If $ x = 6 $: Rent = $ 550 + 25 \times 6 = 700 $.

7. **Check Income for Each Rent**:
   - For $ x = 4 $:
     - Occupied apartments = $ 96 - 3 \times 4 = 84 $.
     - Income = $ 84 \times 650 = 54,600 $. 
     - This matches the goal.

   - For $ x = 6 $:
     - Occupied apartments = $ 96 - 3 \times 6 = 78 $.
     - Income = $ 78 \times 700 = 54,600 $.
     - This also matches the goal.

### Conclusion
The rent could be either:
1. **$650 per month** (with 84 apartments occupied), or
2. **$700 per month** (with 78 apartments occupied).

Do you have any questions, or would you like more details?

### Related Questions:
1. How would the solution change if each $50 increase caused 6 vacancies?
2. What would happen to the rent if the company wanted to achieve an income of $60,000 per month?
3. How many total vacancies would there be at a rent of $675 per apartment?
4. How do changes in vacancy rates impact the overall profit?
5. How can we calculate the rent if the company wants no more than 5 vacancies?

### Tip:
Always check multiple solutions for quadratic equations, as both may be valid for different conditions.

A real-estate firm owns the Parklane Garden Apartments, which consist of 96 apartments. At $550 per month, every apartment can be rented. However, for each $25 per month increase, there will be three vacancies with no possibility of filling them. The firm wants to receive $54,600 per month from rent. What rent should be charged for each apartment?

This problem explores how to find the rent per apartment to achieve a monthly income of $54,600. It involves solving a quadratic equation based on a revenue formula, with each $25 increase in rent causing vacancies. Learn step-by-step how algebra and quadratic equations help solve this.

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