Problem 59 - Prealgebra

Question

Stacy runs a business out of her home making bird houses. Each month she has fixed costs of \(200. In addition, for each bird house she makes, she incurs an additional cost of \)3.00. If her total costs for the month were $296.00, how many bird houses did she make?

Step-by-Step Solution

Answer

Stacy made 32 bird houses.

Step 1: Understand the Problem

Stacy has fixed costs of $200 per month and a variable cost of $3.00 for each bird house she makes. This means that the total monthly cost is the sum of fixed and variable costs. We are tasked with finding out how many bird houses she made if her total monthly cost was $296.

Step 2: Set Up the Equation

Let \( x \) be the number of bird houses she makes. The total cost equation will be the sum of fixed and variable costs: \( 200 + 3x = 296 \).

Step 3: Solve for x

To find \( x \), isolate it on one side of the equation: \( 3x = 296 - 200 \). This simplifies to \( 3x = 96 \). Divide both sides by \( 3 \) to get \( x = \frac{96}{3} \).

Step 4: Simplify the Equation

Calculate \( \frac{96}{3} \) to find the number of bird houses: \( x = 32 \). This means Stacy made 32 bird houses.

Key Concepts

Understanding Fixed Costs

Fixed costs are the unchanging expenses that a business incurs regardless of how much it produces. In Stacy's case, the fixed cost is $200 per month. This cost is consistent and does not fluctuate based on the number of bird houses she makes.

Fixed costs typically include expenses such as rent, insurance, and maintenance fees. These costs are vital for budgeting because they allow businesses to predict how much they will need to spend monthly, regardless of production levels. For any new business, it's essential to identify and calculate fixed costs upfront to understand the minimum financial commitment required for operation.

Knowing your fixed costs can help in setting pricing strategies. If you know what your monthly fixed costs are, you can calculate how much you need to earn each month to at least break even. In Stacy's example, her base cost is $200, whether she makes one birdhouse or a hundred.

• Helps predict cash flow needs.
• Covers necessary expenses, irrespective of output.

Understanding Variable Costs

Variable costs change with the level of production output. For Stacy, every bird house she makes incurs a variable cost of $3.00. This could represent materials like wood, nails, and paint. In contrast to fixed costs, variable costs increase with the production of each additional unit.

Variable costs are crucial for businesses because they directly influence profitability. When planning production, it’s important to understand these costs to make efficient and profitable decisions. For example, if the variable cost is high, Stacy might decide to produce fewer bird houses to control spending.

By managing variable costs, businesses can adjust their production levels efficiently to meet demand and maximize profit. The key lies in finding the balance between fixed and variable costs to sustain a profitable operation. If Stacy's goal is to minimize costs, she might look for cheaper materials or more efficient production methods. Understanding these costs will aid in strategic decision-making.

• Influences cost per unit of product.
• Impacts profitability and pricing structures.

Solving Equations to Find Unknowns

Solving equations is a fundamental skill in math that helps to find unknown values. In business, such as Stacy's birdhouse operation, equations can determine aspects like the number of products needed to be made to achieve a financial goal.

In the problem, we're solving for the number of bird houses made. The equation set up is derived from combining the fixed and variable costs: \[ 200 + 3x = 296 \]Here, \( x \) represents the number of birdhouses. To solve for \( x \), isolate it by moving all other numbers to the opposite side:\[ 3x = 296 - 200 \]This simplifies to:\[ 3x = 96 \]Finally, divide by the coefficient of \( x \):\[ x = \frac{96}{3} \]Solving gives \( x = 32 \), meaning Stacy made 32 bird houses. Understanding how to set up and solve equations helps in making informed decisions and planning effectively. This skill is indispensable in both personal finances and business operations.

• Helps determine production needs.
• Essential for budgeting and financial planning.