Problem 64 - Prealgebra

Question

Stella runs a business out of her home making quilts. Each month she has fixed costs of \(200. In addition, for each quilt she makes, she incurs an additional cost of \)1.75. If her total costs for the month were $280.50, how many quilts did she make?

Step-by-Step Solution

Answer

Stella made 46 quilts.

Step 1: Analyze the Problem

We need to find out how many quilts Stella made given her total cost and the cost structure. This involves isolating the variable that represents the number of quilts.

Step 2: Establish the Equation

Let the number of quilts be \( q \). The total cost equation given the fixed monthly cost and the variable cost per quilt is \( 200 + 1.75q = 280.50 \).

Step 3: Solve for the Variable

Subtract the fixed cost from each side of the equation: \( 1.75q = 280.50 - 200 \), which simplifies to \( 1.75q = 80.50 \).

Step 4: Isolate the Variable

Divide both sides by 1.75 to solve for \( q \): \( q = \frac{80.50}{1.75} \).

Step 5: Calculate the Solution

Perform the division: \( q = 46 \). Therefore, Stella made 46 quilts.

Key Concepts

Equation Solving

Solving an equation is like unraveling a math puzzle. You're trying to find the value of a variable that makes a mathematical statement true. In the problem about Stella and her quilts, we started by establishing an equation based on her costs and the number of quilts she made. The equation we formed was:

• Fixed monthly cost: \(200
• Variable cost per quilt: \)1.75
• Total cost: $280.50

The equation is \[200 + 1.75q = 280.50\]where \(q\) represents the number of quilts. Solving this equation involves logical steps that simplify the problem and break it down to find the value of \(q\). Remember, the key to successful equation solving is to perform the same mathematical operations on both sides to maintain balance.

Variable Isolation

Variable isolation is a vital step in solving equations. It means getting the variable by itself on one side of the equation. Think of it as finding the missing piece in a jigsaw puzzle. In our example, the variable \(q\) is the number of quilts. To isolate \(q\), follow these steps:

• First, subtract the fixed costs from both sides: \(1.75q = 280.50 - 200\).
• This simplifies to \(1.75q = 80.50\).
• Next, divide both sides by the coefficient of \(q\), which is 1.75: \(q = \frac{80.50}{1.75}\).

By isolating \(q\), you transform the equation to a simpler form where the value of \(q\) can be easily calculated. This step-by-step isolation makes the problem easier to handle and solve.

Word Problems

Word problems involve applying mathematical concepts to real-life situations. They can initially seem tricky, but they're just equations in disguise. The key is to translate the text into a mathematical equation. Let's break down the process using Stella's quilt-making business.

• Identify the costs: Fixed and variable.
• Write out the expression: Total cost = Fixed cost + (Cost per quilt \(\times\) Number of quilts).
• Form the equation based on the situation described.

In Stella's case, each quilt added to her total cost and she had fixed expenses each month. By translating her business costs into a formula, we could solve for the number of quilts made. Word problems help us understand how math applies to the world around us, and practicing them can make everyday math tasks much simpler.