Calculating Computer Price Before Tax Heather's $2022.30 Purchase
Understanding Sales Tax and the Original Price
In many purchasing scenarios, particularly when dealing with electronics such as computers, sales tax plays a significant role in the final price. Sales tax, which is a percentage added to the original cost of an item, can sometimes make it challenging to determine the pre-tax price. This article delves into a specific scenario where Heather paid $2,022.30 for a computer, a price that includes a 7% sales tax. Our primary goal is to figure out the original cost of the computer before the sales tax was applied. We will explore various mathematical equations that can be used to accurately calculate this pre-tax price. Understanding these calculations is crucial not only for academic purposes but also for real-world financial literacy, allowing consumers to make informed decisions about their purchases.
At the heart of this problem lies the relationship between the original price, the sales tax rate, and the final price paid. The original price represents the cost of the computer before any tax is added. The sales tax rate, in this case, is 7%, which means that 7% of the original price is added to the original price to arrive at the final price. The final price, which Heather paid, is $2,022.30. To unravel this, we will use algebraic equations, a powerful tool in mathematics for solving problems involving unknown quantities. By setting up the correct equation, we can isolate the original price and determine its value. This involves understanding how to represent percentages as decimals and how to manipulate equations to solve for the unknown variable. Let’s dive deeper into the process of setting up and solving these equations to find the original price of the computer.
Furthermore, it’s essential to grasp the concept of percentages and their conversion to decimals. A percentage is essentially a fraction out of 100, so 7% is equivalent to 7/100, which as a decimal is 0.07. This decimal representation is critical when we set up our equation because we will be multiplying the original price by this decimal to find the amount of sales tax. The total amount Heather paid includes both the original price of the computer and the 7% sales tax on that price. Therefore, if we let x represent the original price of the computer, the sales tax amount would be 0.07 times x, or 0.07x. The equation we form must reflect the sum of these two components equaling the total amount paid. This step-by-step breakdown is designed to ensure that the underlying mathematical principles are clear, setting a solid foundation for constructing the appropriate equation.
Defining Variables: x as the Cost and y as the Total
In mathematical problem-solving, defining variables is a critical first step. This process involves assigning letters or symbols to represent the unknown quantities we are trying to determine. In the context of Heather’s computer purchase, we are given a specific instruction: let x represent the cost of the computer before tax and y represent the total amount paid, which includes the sales tax. Defining these variables helps us to translate the word problem into a mathematical equation that we can then solve. The clear definition of variables ensures that everyone working on the problem is using the same understanding and notations, which is essential for accuracy and consistency in the solution. This initial step is foundational, guiding the subsequent steps in formulating the equation and ultimately finding the solution. The choice of letters is arbitrary, but it is common practice to use letters that are easy to remember and relate to the quantities they represent.
By defining x as the cost of the computer, we are establishing the unknown value that we aim to find. The original cost is the base price before any additional fees or taxes are applied. This variable is crucial because the sales tax is calculated as a percentage of this original cost. The total amount paid, represented by y, is the final price that Heather paid, which includes both the original cost and the sales tax. In this case, y is given as $2,022.30. The relationship between x and y is what we need to capture in an equation. This equation will mathematically express how the original cost x and the sales tax combine to give the total amount paid y. The clarity in defining these variables sets the stage for creating a precise and solvable equation. Without this step, we would be working with an unclear representation of the quantities involved, making the solution process much more complex.
Moreover, understanding the relationship between x and y is key to setting up the correct equation. The total amount paid (y) is the sum of the original cost (x) and the sales tax. The sales tax, as we discussed earlier, is 7% of the original cost. Therefore, we can express the sales tax amount as 0.07x. Adding this to the original cost gives us the total amount paid. This relationship can be written as an equation, which will allow us to solve for x, the original cost of the computer. The act of defining variables helps to clarify this relationship, making it easier to translate the problem into a solvable form. This methodical approach is a cornerstone of effective problem-solving in mathematics and other disciplines.
Formulating Equations to Determine the Price
The formulation of equations is the crux of solving mathematical word problems. Now that we have defined our variables (x as the original cost of the computer and y as the total amount paid) and understood the relationships between them, we can create an equation that accurately represents the situation. The equation needs to express how the original cost plus the sales tax equals the total amount paid. As we’ve established, the sales tax is 7% of the original cost, which can be written as 0.07x. Therefore, the equation should include terms for the original cost (x), the sales tax (0.07x), and the total amount paid (y). This equation will serve as a mathematical model of the problem, allowing us to use algebraic techniques to solve for the unknown variable, which in this case is the original cost of the computer.
The equation that represents this scenario is x + 0.07x = y. This equation states that the original cost of the computer (x) plus the 7% sales tax on the computer (0.07x) equals the total amount paid (y). This is a linear equation in one variable, which is a straightforward type of equation to solve. The left side of the equation combines the original cost and the sales tax, while the right side represents the total amount Heather paid. Now, we know that Heather paid $2,022.30, so we can substitute y with this value. This substitution gives us the equation x + 0.07x = $2,022.30. This refined equation provides a concrete mathematical statement that we can use to find the original cost of the computer.
To solve for x, we first need to combine like terms on the left side of the equation. The terms x and 0.07x are like terms because they both contain the variable x. Combining these terms involves adding their coefficients. The coefficient of x is 1 (since x is the same as 1x), and the coefficient of 0.07x is 0.07. Adding these coefficients gives us 1 + 0.07 = 1.07. Thus, the equation simplifies to 1.07x = $2,022.30. This simplified equation is now easier to solve for x. We can solve for x by dividing both sides of the equation by 1.07. This isolates x on one side of the equation, giving us the original cost of the computer. The final step in solving the equation will provide the numerical value of x, answering the question of the computer’s original price.
Solving for x: Determining the Original Cost
Having formulated the equation 1.07x = $2,022.30, solving for x is the next critical step in finding the original cost of the computer. To isolate x, we need to perform an operation that will undo the multiplication by 1.07. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 1.07. This maintains the equality of the equation while moving us closer to finding the value of x. This is a fundamental principle of algebraic manipulation: whatever operation is performed on one side of the equation must also be performed on the other side to keep the equation balanced.
Dividing both sides of the equation 1.07x = $2,022.30 by 1.07 gives us x = $2,022.30 / 1.07. Performing this division yields x ≈ $1,890. This result means that the original cost of the computer before the 7% sales tax was applied is approximately $1,890. The division operation effectively separates the original cost from the added sales tax, allowing us to pinpoint the computer's pre-tax price. This calculation demonstrates the power of algebraic equations in solving real-world financial problems.
Therefore, we have successfully found the original cost of the computer by using the equation we formulated. This process highlights the importance of understanding the relationship between the original price, the sales tax, and the total price. By setting up the equation x + 0.07x = $2,022.30 and solving for x, we were able to determine the computer's original price. This methodical approach to problem-solving is a valuable skill, applicable not only in mathematics but also in various practical situations. The ability to break down a complex problem into smaller, manageable steps is key to achieving a solution.
Validating the Solution and Final Answer
After solving for x, validating the solution is an essential step to ensure accuracy. In our case, we found that the original cost of the computer, x, is approximately $1,890. To validate this, we need to check if adding the 7% sales tax to $1,890 results in the total amount Heather paid, which is $2,022.30. This validation process confirms that our calculations are correct and that the solution we obtained is indeed the correct one. Validation is a critical part of the problem-solving process, as it helps to catch any potential errors in calculation or formulation.
To validate our solution, we first calculate the sales tax amount by multiplying the original cost, $1,890, by the sales tax rate, 7% (or 0.07). This gives us 0.07 * $1,890 = $132.30. This is the amount of sales tax that was added to the original cost. Next, we add this sales tax amount to the original cost: $1,890 + $132.30 = $2,022.30. This result matches the total amount Heather paid for the computer, which confirms that our solution for x is accurate. The validation process ensures that the answer is consistent with the information given in the problem.
In conclusion, the original cost of the computer before the 7% sales tax was applied is approximately $1,890. We arrived at this solution by defining variables, formulating an equation, solving for x, and validating our result. This comprehensive process demonstrates a thorough approach to problem-solving, ensuring both accuracy and understanding. The ability to solve such problems is not only valuable in academic settings but also in real-life scenarios involving financial calculations and decision-making. The steps taken to solve this problem can be applied to various situations where it is necessary to calculate prices, taxes, or discounts.
