A machine company has an incentive plan for its salespeople. For each machine that a salesperson sells, the commission is $50. The commission for every machine will increase by $0.05 for each machine sold over 500. For example, the commission on each of 502 machine sold is $50.10. How many machines must a salesperson sell in order to earn $33.000?

We can break this problem down into two parts. First, we need to find the commission for the initial 500 machines, and then we calculate the commission for machines sold beyond 500.

### Step 1: Define Variables
- Let $ n $ be the total number of machines sold.
- For the first 500 machines, the commission is fixed at $50 per machine.
- For any machine sold beyond 500, the commission increases by $0.05 per machine.

### Step 2: Calculate Total Commission for the First 500 Machines
If the salesperson sells 500 machines, the total commission for these machines is:
\[
500 \times 50 = 25,000
\]
So, the commission for the first 500 machines is $25,000.

### Step 3: Commission for Machines Sold Beyond 500
For each machine sold beyond 500, the commission increases by $0.05 per machine. That means:
- The commission for the 501st machine is $ 50 + 0.05 = 50.05 $
- The commission for the 502nd machine is $ 50 + 0.10 = 50.10 $
- And so on...

If the salesperson sells $ n - 500 $ machines beyond 500, we need to sum the commissions for these machines. The total commission for machines sold beyond 500 is a sum of an arithmetic sequence with:
- First term $ a = 50.05 $
- Common difference $ d = 0.05 $
- Number of terms $ n - 500 $

The formula for the sum of an arithmetic sequence is:
\[
S = \frac{m}{2} \times (2a + (m-1)d)
\]
where $ m = n - 500 $, $ a = 50.05 $, and $ d = 0.05 $.

### Step 4: Total Earnings
The total earnings is the sum of the commission for the first 500 machines and the sum for the machines beyond 500. This total must equal $33,000. Hence, we write the equation:
\[
25,000 + S = 33,000
\]
Solving this equation will give us the value of $ n $, the total number of machines sold.

I'll calculate this next.

A machine company has an incentive plan for its salespeople. For each machine that a salesperson sells, the commission is $50. The commission for every machine will increase by $0.05 for each machine sold over 500. For example, the commission on each of 502 machines sold is $50.10. How many machines must a salesperson sell in order to earn $33,000?

Solve a sales commission problem where the commission increases with each machine sold after 500. Learn how to apply arithmetic sequence formulas and calculate the total number of machines sold to reach a commission of $33,000. This problem covers key concepts like arithmetic progression and linear equations, suitable for students in Grades 9-12.

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