To solve for the base commission, let's break down the problem step by step.

Step 1: Breakdown of Sales and Commission Structure

1. First $75,000 in sales: Earns a base commission.
2. Next $75,000 in sales (from $75,001 to $150,000): Earns base + 2%.
3. Sales above $150,000 (from $150,001 to $200,000): Earns base + 4%.

Step 2: Express the Total Earnings in Terms of the Base Commission

Let the base commission be $B$.

• For the first $75,000 in sales, the earnings are $B$.
• For the next $75,000 in sales (i.e., $75,001 to $150,000), the earnings are $B + 0.02 \times 75,000$.
• For the last $50,000 in sales (i.e., $150,001 to $200,000), the earnings are $B + 0.04 \times 50,000$.

Step 3: Set Up the Equation

The total gross earnings are given as $7,500. Therefore, we set up the equation: $B + (B + 0.02 \times 75,000) + (B + 0.04 \times 50,000) = 7,500$

Step 4: Simplify the Equation

First, calculate the additional earnings:

• $0.02 \times 75,000 = 1,500$
• $0.04 \times 50,000 = 2,000$

So, the equation becomes: $B + (B + 1,500) + (B + 2,000) = 7,500$

Simplify: $3B + 3,500 = 7,500$

Step 5: Solve for $B$

Subtract 3,500 from both sides: $3B = 4,000$

Divide by 3: $B = \frac{4,000}{3} \approx 1,333.33$

So, the base commission $B$ is approximately $1,333.33.

Verification

Let's verify by plugging $B = 1,333.33$ back into the earnings equation:

• For the first $75,000: $1,333.33
• For the next $75,000: $1,333.33 + $1,500 = $2,833.33
• For the last $50,000: $1,333.33 + $2,000 = $3,333.33

Total earnings: $1,333.33 + $2,833.33 + $3,333.33 = $7,500

The calculations are correct, confirming that the base commission is indeed $1,333.33.

Would you like more details or have any further questions?

Here are 5 related questions for you:

1. What would be the total gross earnings if the base commission was $1,500?
2. How would the base commission change if the percentage on the last $50,000 was 3% instead of 4%?
3. What if the base commission applied to the first $100,000 instead of $75,000?
4. How would the structure change if there was a flat 2% commission on all sales?
5. What is the impact on gross earnings if sales are increased to $250,000?

Tip: When solving problems with piecewise functions or tiered structures, break down the problem into its individual parts before combining them. This approach simplifies the analysis and ensures accuracy.