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.