Problem 46 - Algebra 2

Question

Mrs. Lucas earns a salary of \(\$ 34,000\) per year plus 1.5\(\%\) commission on her sales. If the average price of a car she sells is \(\$ 30,500\) , about how many cars must she sell to make an annual income of at least \(\$ 50,000 ?\) Solve the inequality and interpret the solution.

Step-by-Step Solution

Answer

Mrs. Lucas must sell at least 35 cars to earn at least $50,000 annually.

Step 1: Understand the Problem

Mrs. Lucas earns a base salary of \( \\(34,000 \) annually, plus a commission of 1.5\(\%\) on each car sold. We need to determine how many cars she must sell to earn at least \( \\)50,000 \) annually.

Step 2: Convert Percentage to Decimal

To calculate the commission she earns on a car sale, convert the percentage to a decimal. Thus, 1.5\(\%\) becomes 0.015.

Step 3: Calculate Commission per Car

The commission earned per car is given by multiplying the commission rate by the price of one car. This is \( 0.015 \times 30,500 \).

Step 4: Calculate Total Income

Let \( x \) be the unknown number of cars she must sell. Her total income is the sum of her base salary and the commission. This can be represented by the equation: \( 34,000 + 0.015 \times 30,500 \times x = 50,000 \).

Step 5: Set Up Inequality

Mrs. Lucas wants to earn at least \( \$50,000 \). Therefore, set up the inequality: \( 34,000 + (0.015 \times 30,500) \times x \geq 50,000 \).

Step 6: Solve the Inequality

First, calculate the commission per car: \( 0.015 \times 30,500 = 457.5 \). Substitute this into the inequality: \( 34,000 + 457.5x \geq 50,000 \). Then, simplify and solve for \( x \):1. \( 457.5x \geq 50,000 - 34,000 \).2. \( 457.5x \geq 16,000 \).3. Divide both sides by 457.5: \( x \geq \frac{16,000}{457.5} \approx 34.97 \).4. Since she can't sell a fraction of a car, round up to 35 cars.

Step 7: Interpret the Solution

To earn at least \( \$50,000 \) in a year, Mrs. Lucas must sell at least 35 cars.

Key Concepts

Commission calculation

When someone earns a commission, it means they receive extra money based on sales or performance. In Mrs. Lucas's case, her commission is calculated at a rate of 1.5% for each car she sells. To work with percentages in mathematical calculations, it's essential to convert this percentage into a decimal before using it in formulas. Here’s how her commission calculation works:

• Convert 1.5% to a decimal by dividing by 100, resulting in 0.015.
• Multiply this decimal by the average car price she sells, which is \( 30,500 \), to find the commission per car.
• This gives us \( 0.015 \times 30,500 = 457.5 \) dollars as commission for each car.

This commission amount is then added to her base salary to reach her total income each year.

Income calculation

Mrs. Lucas's total income comes from two parts: her base salary and the commission she makes from selling cars. To find out how much she earns in a year, you will need to add these two income sources together:

• Base Salary: She has a fixed annual salary of \( \\( 34,000 \).
• Commission: We previously calculated that she earns \( 457.5 \) dollars in commission per car sold.

To find her total income if she sells \( x \) cars:

• Multiply the commission per car by the number of cars \( x \), giving \( 457.5x \).
• Add the base salary to this product: \( 34,000 + 457.5x \).

This formula gives us her total annual income, which helps us determine how many cars she needs to sell to reach at least \( \\) 50,000 \).

Solving inequalities

Inequalities are just like equations, but instead of an equality sign, we use inequality signs such as \( \frac{\text{\gt}}{\underline{\phantom{xx}}} \) or \( \frac{\text{\lt}}{\underline{\phantom{xx}}} \). In this problem, we want Mrs. Lucas's income to be at least \( \$ 50,000 \), which means she should earn no less than this amount. This leads to the inequality:

• Set up the inequality: \( 34,000 + 457.5x \geq 50,000 \).
• To solve it, first subtract her base salary from the desired income: \( 457.5x \geq 50,000 - 34,000 \).
• This simplifies to \( 457.5x \geq 16,000 \).
• Next, divide both sides by 457.5 to isolate \( x \): \( x \geq \frac{16,000}{457.5} \).
• After calculation: \( x \approx 34.97 \).
• Since it's not possible to sell a fraction of a car, round up to the nearest whole number: 35 cars.

Thus, Mrs. Lucas needs to sell at least 35 cars to meet her income goal, according to the inequality solution. Solving inequalities helps to understand how real-life variables can interact to meet certain conditions, providing clarity on financial or strategic planning.