Problem 70 - Prealgebra

Question

A stationary store has engineering templates on sale for \(2.50 apiece. A business purchases an unknown number of these and the total cost of their purchase is \)60.00. How many were purchased?

Step-by-Step Solution

Answer

24 templates were purchased.

Step 1: Identify Known Values

We know that each engineering template costs $2.50 and the total cost of the purchase is $60.00. We want to find out how many templates were purchased.

Step 2: Set Up the Equation

Let the number of templates be represented by the variable \( x \). The total cost for \( x \) templates is given by the equation \( 2.50x = 60.00 \).

Step 3: Solve the Equation

To find \( x \), divide both sides of the equation by 2.50:\[ x = \frac{60.00}{2.50} \]

Step 4: Calculate the Division

Perform the division:\[ x = 24 \]

Key Concepts

Linear Equations

A linear equation is an essential concept in algebra that helps in finding an unknown variable efficiently. When dealing with algebra word problems like the one in our example, we're often tasked with setting up such equations to find the solution.
Linear equations have the form:

• \[ ax + b = c \]

In this form, \( a \), \( b \), and \( c \) are known values, while \( x \) is the unknown variable we're trying to solve. In the context of our problem, \( 2.50x = 60.00 \) is the linear equation. Here, \( a = 2.50 \) is the cost per engineering template, \( b = 0 \) as there is no extra fee, and \( c = 60.00 \) is the total cost.
We aim to solve such an equation by isolating \( x \), which allows us to find how many units, or items in this case, were purchased.

Problem-Solving Steps

To solve word problems involving linear equations, it is crucial to follow specific problem-solving steps. These steps guide us in organizing the given information and systematically applying the mathematical operations needed.
Let's run through the steps as applied to our stationary store problem:

• Identify Known Values: Determine the fixed values given in the problem. Here, we know each template costs \(2.50\) and the purchase total is \(60.00\).
• Set Up the Equation: Use the known values to represent the situation as a linear equation. We do this by letting \(x\) denote the number of templates, leading us to \(2.50x = 60.00\).
• Solve the Equation: Isolate the variable \(x\) by performing arithmetic operations. Divide both sides by \(2.50\) to find \(x\).
• Calculate the Result: Perform the calculations needed, such as division, to get the specific value for \(x\). In our exercise, dividing \(60.00\) by \(2.50\) yields \(x = 24\).
• Verify the Solution: Optionally, plug \(x\) back into the context of the problem to ensure the solution makes sense. Here, multiply \(24\) by \(2.50\) to confirm it equals \(60.00\).

By neatly following these steps, solving algebra word problems becomes more manageable and less error-prone.

Unit Cost Calculation

Unit cost calculation is a common task in real-world scenarios where one needs to find out the cost associated with a specific number of units. This is particularly useful in business settings where budgeting and cost estimation are crucial.
For our problem, the unit cost is \(2.50\) per engineering template. The task was to determine how many templates could be bought within a total cost of \(60.00\). This involves dividing the total expenditure by the cost per unit. The formula looks like this:

• \[ x = \frac{\text{Total Cost}}{\text{Unit Cost}} \]

When applied to the exercise:

• \[ x = \frac{60.00}{2.50} = 24 \]

This means that for \(60.00\), 24 templates can be purchased. Understanding unit cost calculations helps in many situations beyond this example, such as planning how many supplies are needed or predicting expense reports.