Problem 328 - Elementary Algebra

Question

In the following exercises, solve. Melissa makes necklaces and sells them online. She charges \(\$ 88\) per necklace. Her monthly expenses are \(\$ 3745 .\) How many necklaces must she sell if she wants to make a profit of at least \(\$ 1,650 ?\)

Step-by-Step Solution

Answer

Melissa must sell at least 62 necklaces.

Step 1: Define Variables

Let the number of necklaces Melissa needs to sell be represented by the variable \( n \).

Step 2: Set Up the Revenue Equation

The revenue from selling \( n \) necklaces is given by multiplying the number of necklaces by the price per necklace: \( 88n \).

Step 3: Set Up the Profit Equation

The profit Melissa makes is the revenue minus her monthly expenses. Write the profit equation: \( \text{Profit} = 88n - 3745 \).

Step 4: Set Profit to at Least 1650

Melissa wants to make at least \(1650 \) in profit. Set up the inequality: \( 88n - 3745 \geq 1650 \).

Step 5: Solve the Inequality

Add \( 3745 \) to both sides of the inequality to isolate the term with \( n \): \( 88n \geq 1650 + 3745 \) Simplify the right side: \( 88n \geq 5395 \). Divide both sides by \( 88 \) to solve for \( n \): \( n \geq 61.31 \).

Step 6: Interpret the Result

Since Melissa cannot sell a fraction of a necklace, she must sell at least 62 necklaces to make a profit of at least \(1650 \).

Key Concepts

Understanding Inequalities in Profit Calculation

In any business scenario, you often need to determine minimum or maximum thresholds. This is where inequalities come into play. In this exercise, Melissa wants to ensure her profit is at least \(1,650. To express this, we use the inequality \( 88n - 3745 \geq 1650 \). This inequality means the left side (revenue minus expenses) must be greater than or equal to the right-side value (\)1,650). This is an example of a simple linear inequality, where we perform algebraic operations to solve it.

Profit Calculation Made Simple

Profit is essentially the money left over after all expenses are paid. In this situation, the profit formula is \( \text{Profit} = \text{Revenue} - \text{Expenses} \). For Melissa, the revenue comes from selling necklaces, and her expenses are the costs she incurs each month. The equation becomes \( 88n - 3745 \), where \( 88n \) represents the total revenue from selling \( n \) necklaces at \(88 each, and her fixed expenses are \)3,745. To ensure a profit of at least $1,650, the expression must fulfil the inequality \( 88n - 3745 \geq 1650 \).

Linear Equations in Business Contexts

Linear equations are fundamental in various calculations, including business and finance. For example, the equation \( 88n - 3745 = 1650 \) can help us determine exactly how many items must be sold to achieve a specific profit. By isolating the variable \( n \), we solve for the number of necklaces needed. Utilizing linear equations, we simplify complex problems into manageable steps: calculate total revenue, subtract expenses, and compare with the desired profit.

Variable Representation in Algebra

Choosing the right variables simplifies problem-solving. In this context, let \( n \) represent the number of necklaces Melissa sells. This approach allows for clear, concise equations and eases manipulation during calculations. For example, \( 88 \times n \) directly translates to the revenue from \( n \) necklaces. This clear representation leads to easily forming equations and solving for the desired variable, yielding meaningful results like determining the minimum necklaces Melissa needs to sell.