Let's break down the problem step by step.

Step 1: Understand the commission structure.

Marissa's commission structure is as follows:

1. 3.7% on the first $300,000 of the selling price.
2. 3.2% on the next $250,000 of the selling price.
3. 2.7% on any remaining amount.

If the property is sold at auction, an extra fee of $4,750 is charged.

Step 2: Define the selling price.

Let the selling price be $x$.

Step 3: Calculate the commission based on the selling price.

• For the first $300,000: $\text{Commission for first } 300,000 = 0.037 \times 300,000 = 11,100$
• For the next $250,000 (i.e., the amount between $300,000 and $550,000): $\text{Commission for next } 250,000 = 0.032 \times 250,000 = 8,000$
• For any remaining amount above $550,000 (i.e., $x - 550,000$): $\text{Remaining amount} = x - 550,000$ $\text{Commission on remaining amount} = 0.027 \times (x - 550,000)$

Step 4: Express the total commission (excluding the auction fee).

The total commission Marissa receives before adding the auction fee is: $\text{Total commission} = 11,100 + 8,000 + 0.027 \times (x - 550,000)$

Step 5: Include the auction fee.

Including the auction fee of $4,750, the total amount she receives is: $\text{Total amount} = 11,100 + 8,000 + 0.027 \times (x - 550,000) + 4,750$ This is equal to $37,500, so we can set up the equation: $11,100 + 8,000 + 0.027 \times (x - 550,000) + 4,750 = 37,500$

Step 6: Solve for $x$.

Simplify and solve the equation: $23,850 + 0.027(x - 550,000) = 37,500$

Subtract 23,850 from both sides: $0.027(x - 550,000) = 13,650$

Now, divide by 0.027: $x - 550,000 = \frac{13,650}{0.027} = 505,555.56$

Finally, add 550,000 to both sides: $x = 550,000 + 505,555.56 = 1,055,555.56$

Final Answer:

The selling price of the property is approximately $1,055,555.56.

Would you like a detailed breakdown of the calculations or have any questions? Here are 5 related questions to deepen your understanding:

1. How would the total commission change if the property was sold for $1,200,000?
2. What would Marissa's total commission be if the property sold for $400,000 without the auction fee?
3. How does the commission rate affect the total earnings as the selling price increases?
4. How would Marissa's commission structure affect properties that are sold just above $550,000?
5. How can we verify if Marissa's commission was calculated correctly for a different selling price?

Tip: When setting up equations, break the problem into smaller parts to simplify the calculations and reduce errors.