Problem 55 - Algebra 2

Question

SALES Sales associates at Electronics Unlimited earn \(\$ 8\) an hour plus a 4\(\%\) commission on the merchandise they sell. Write a function to describe their income, and find how much merchandise they must sell in order to earn \(\$ 500\) in a 40 -hour week. (Lesson \(7-2 )\)

Step-by-Step Solution

Answer

Sell $4500 worth of merchandise to earn $500 in a 40-hour week.

Step 1: Understanding the Problem

We need to find a function that represents the total income of a sales associate based on hours worked and merchandise sold. Then we will find how much merchandise needs to be sold to reach a specific income goal.

Step 2: Defining the Function

Sales associates earn a base pay and a commission. The base pay is computed as $8 per hour for a 40-hour week: \[ Base\ Income = 8 \times 40 = 320 \,\text{dollars} \] They also earn a 4% commission on merchandise sold, so the income function is \[ I(x) = 320 + 0.04x \] where \( x \) is the dollar amount of merchandise sold.

Step 3: Setting the Income Goal

We need to find \( x \) when the total income is $500. Therefore, the income equation is \[ 320 + 0.04x = 500 \]

Step 4: Solving for Merchandise Sold

Subtract 320 from both sides: \[ 0.04x = 180 \]Now, divide both sides by 0.04 to solve for \( x \): \[ x = \frac{180}{0.04} = 4500 \] So, $4500 worth of merchandise needs to be sold.

Key Concepts

Commission Calculation

To comprehend how commission calculation works in this context, it's essential to understand both its meaning and its role in income formulation. Commission is a percentage-based payment that sales associates receive based on the sales they make. It acts as an incentive for them to sell more products.

In the given problem, the commission rate is 4%. This means that for every dollar's worth of merchandise sold, the sales associate earns an additional 4 cents. Mathematically, this is calculated as:

• Commission earned = Commission rate × Merchandise sold
• In formula, this is represented as 0.04x, where x is the total sales.

By adding this commission to the base income, we create the complete income function for the sales associate, which helps determine earnings based on their sales performance.

Base Pay Computation

Base pay is the fixed hourly wage that a sales associate earns regardless of sales performance. In this case, the associate earns $8 per hour.

For a standard 40-hour work week, the base pay can be computed with a simple multiplication:

• Base Pay = Hourly Wage × Hours Worked
• For this scenario: Base Pay = 8 × 40 = 320 dollars

This calculation forms the foundational amount of the associate’s weekly earnings. It's crucial for workers to understand their base pay, as it both ensures financial stability and provides a benchmark for how much additional earning can be made from selling merchandise, via commissions.

Merchandise Sales Goal

Determining the merchandise sales goal involves solving for the amount of merchandise that needs to be sold to achieve a specific income. In this example, the target income is $500 for a 40-hour work week.

The total income combines base pay and commission from sales (x dollars of merchandise), expressed as follows:

• Total Income = Base Income + Commission
• Using the formula: 320 + 0.04x

To reach the goal of $500, you set the equation to solve for x:

• 320 + 0.04x = 500
• By solving, we find x = 4500, meaning $4500 in merchandise sales is needed.

Understanding this promotes goal setting which can boost performance as it quantifies what sales targets must be met for desired earnings.