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The image describes a problem where the manager of a publishing company plans to distribute bonuses to its sales representatives based on three tiers:

- The top **11%** receive a **$28,000** bonus.
- The next **34%** receive a **$10,000** bonus.
- The next **9%** receive a **$5,000** bonus.

The company has **260 sales representatives** in total.

The questions ask for:

1. The **expected bonus per sales representative**.
2. The **total expected bonus** that the company will pay to all sales representatives combined.

### Solution:

#### 1. Expected Bonus Per Sales Representative

The expected bonus for an individual sales representative can be calculated using the formula for expected value. We multiply the percentage of representatives in each group by the bonus they will receive.

Let's break it down:
- **11%** of 260 representatives receive **$28,000**:  
  $ 0.11 \times 260 = 28.6 $ representatives receive $28,000.
- **34%** of 260 representatives receive **$10,000**:  
  $ 0.34 \times 260 = 88.4 $ representatives receive $10,000.
- **9%** of 260 representatives receive **$5,000**:  
  $ 0.09 \times 260 = 23.4 $ representatives receive $5,000.

The expected bonus is:
\[
\text{Expected Bonus} = \left( \frac{28.6}{260} \times 28,000 \right) + \left( \frac{88.4}{260} \times 10,000 \right) + \left( \frac{23.4}{260} \times 5,000 \right)
\]
Let me calculate the result.

#### 2. Total Expected Bonus Paid by the Company

The total bonus paid by the company will be:
\[
\text{Total Expected Bonus} = (28.6 \times 28,000) + (88.4 \times 10,000) + (23.4 \times 5,000)
\]
I will calculate these now.### Results:
1. The **expected bonus per sales representative** is **$6,930**.
2. The **total expected bonus** that the company will pay to all sales representatives is **$1,801,800**.

Let me know if you'd like more details or have any questions!

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Here are five related questions:
1. How would the expected bonus change if the number of representatives increased?
2. What would happen if the bonus amounts for each tier were adjusted?
3. How does the expected bonus per representative compare if only two bonus tiers were used?
4. What if the top percentage of representatives was increased to 20%?
5. How would reducing the company's total budget for bonuses impact each representative's bonus?

**Tip:** When calculating expected values, always multiply the probability (or percentage) by the associated value and sum the results for all groups.

The manager of a publishing company plans to give a $28,000 bonus to the top 11%, $10,000 to the next 34%, and $5,000 to the next 9% of sales representatives. If the company has 260 representatives, what is the expected bonus per representative and the total expected bonus for all employees combined?

Solve the problem of calculating the expected bonus for sales representatives using weighted averages and percentages. The problem involves distributing different bonuses to tiers of employees based on performance percentages. This guide is suitable for Grades 10-12.

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