Problem 79 - A Graphical Approach to College Algebra

Question

From 2010 to \(2012,\) the average selling price of tablets decreased by \(30 \% .\) This percent reduction amounted in a decrease of 195 dollars. Find the average selling price of tablets in 2010 and in 2012.

Step-by-Step Solution

Answer

The price was $650 in 2010 and $455 in 2012.

Step 1: Understand the Percent Decrease

The average selling price of tablets decreased by 30%. This means that if the original price in 2010 is represented as \( P \), the drop in price would be represented as \( 0.3 \times P \). We know this amount is equal to 195 dollars.

Step 2: Set Up the Equation

Using the information provided, we can establish the equation for the decrease: \[ 0.3 \times P = 195 \]. This equation connects the percentage decrease directly to the dollar amount decrease.

Step 3: Solve for Initial Price in 2010

To find \( P \), divide both sides of the equation by 0.3: \[ P = \frac{195}{0.3} \]. This calculation will give us the average selling price of tablets in 2010.

Step 4: Calculate the Price in 2010

Perform the division: \[ P = 650 \]. Thus, the average selling price of tablets in 2010 was 650 dollars.

Step 5: Calculate the Price in 2012

Since the price decreased by 195 dollars, subtract 195 from the 2010 price to find the 2012 price: \[ 650 - 195 = 455 \]. The average selling price in 2012 is 455 dollars.

Key Concepts

Understanding Average Selling Price

When we talk about the "average selling price," we're referring to the typical price at which products, like tablets, are sold over a specific period. This concept is crucial for businesses since it helps them determine how their products are positioned in the market.

To calculate the average selling price, businesses often do the following:

• Sum up all the sales revenue obtained from selling the product.
• Divide this total revenue by the number of units sold.

This formula gives insight into pricing trends and future strategies. For example, if tablets were selling at an average of $650 in 2010, this sets a baseline for evaluating performance in subsequent years, like 2012.

Mastering Equation Solving

Equation solving is a foundational skill in mathematics and often involves finding the value of an unknown variable. In our exercise, equation solving is applied to find the initial average selling price of tablets.

The step-by-step breakdown is as follows:

• Identify the unknown variable: Here, it’s the average selling price in 2010, represented as \( P \).
• Set up the equation: With a 30% decrease resulting in a $195 dollar drop, we express this as \( 0.3 \times P = 195 \).

Next, solve for \( P \): Divide both sides of the equation by 0.3:\[P = \frac{195}{0.3}\] This calculation helps pinpoint the 2010 price, demonstrating how algebra can be used to solve real-world problems.

Implementing Mathematical Modeling

Mathematical modeling involves using equations and calculations to represent real-world scenarios. This is a powerful way to predict and analyze financial and business performances.

In this exercise, mathematical modeling aids in understanding how a percentage decrease affects the selling prices from one year to another. The model we created involves:

• Defining a real-world problem in mathematical terms, e.g., the price drop of tablets.
• Using equations, like \( 0.3 \times P = 195 \), to link percentage decreases to tangible dollar amounts.

By calculating changes, such as subtracting the \(195 decrease from the original \)650 price, we employ modeling to get a clear financial picture, offering valuable insights into market behavior and pricing strategies.