A sales organization pays a base commission on the first $75,000 in sales, base + 2% on the next $75,000 in sales, and
base + 4% on anything above. What is the base commission if an employee received total gross earnings of $7,500 on
$200,000 in sales?

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.

Learn how to calculate the base commission on $200,000 in sales using a tiered structure. This problem involves understanding piecewise functions and percentage calculations for sales commissions. Suitable for advanced high school students and beyond.

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