Problem 91 - A Graphical Approach to Precalculus with Limits

Question

Price of Smartphones From 2014 to \(2016,\) the average selling price of smartphones decreased by \(10 \% .\) This percent reduction amounted in a decrease of \(\$ 31 .\) Find the average selling price of smartphones in 2014 and in 2016 .

Step-by-Step Solution

Answer

In 2014, the price was $310; in 2016, it was $279.

Step 1: Define the Variables

Let \( x \) be the average selling price of smartphones in 2014. According to the problem, there was a 10% reduction, which is represented as 0.10. Therefore, the decrease in price is written as \( 0.10x = 31 \).

Step 2: Set Up the Equation

We know from Step 1 that the decrease in price over the two years was \( 31 \). So the equation we have is \( 0.10x = 31 \). We will use this equation to solve for \( x \).

Step 3: Solve the Equation

To find \( x \), divide both sides of the equation by 0.10: \[ x = \frac{31}{0.10} = 310 \]Thus, the average selling price in 2014 was \( 310 \) dollars.

Step 4: Find the Average Selling Price in 2016

We know the price decreased by \( 31 \) dollars from 2014 to 2016. To find the average selling price in 2016, subtract \( 31 \) from the 2014 price:\[ 310 - 31 = 279 \]So, the average selling price of smartphones in 2016 was \( 279 \) dollars.

Key Concepts

Understanding Percentage Decrease

When dealing with financial or quantitative data, understanding percentage decrease is crucial. Percentage decrease shows how much a quantity has reduced relative to its original amount. To calculate this, we use the formula:

• First, subtract the new value from the original value to find the decrease amount.
• Then, divide this decrease by the original value.
• Finally, multiply by 100 to get the percentage decrease.

In the context of our smartphone price exercise, the selling price decreased by 10%, which is expressed as a decimal by dividing 10 by 100, resulting in 0.10. This means that the price went down by 10% of its original value, \(x\). Therefore, the reduction is noted as \(0.10x\), indicating 10% of the price in 2014 was lost. Understanding these steps allows us to interpret how figures change in contexts like pricing and finance.

Introduction to Variable Representation

Variables are symbols or letters used to represent unknown or changeable values in mathematics. They let us create expressions and equations that model real-world situations.
For instance, in our problem, we use the variable \(x\) to symbolize the unknown initial smartphone average selling price in 2014. This approach creates a relationship between the percentage decrease and the actual monetary decrease. It translates the statement into an equation:

• The variable \(x\) stands for the original price.
• The expression \(0.10x\) represents the decrease caused by the 10% reduction.

Using variables allows for flexibility and understanding, making problem-solving systematic and clear. This skill is not just limited to financial math but widely applicable to various fields involving measurements and predictions.

Basic Techniques of Equation Solving

Equation solving is a fundamental exercise in mathematics that empowers us to find unknown values by manipulating equations strategically. In our example, the task is to isolate the variable \(x\), representing the original smartphone price. The equation is given by: \(0.10x = 31\).
To solve for \(x\), you must:

• Recognize that \(0.10x\) indicates 10% of \(x\), equaling 31.
• Rearrange the equation to solve for \(x\) by dividing both sides by 0.10.

Thus, the equation \(x = \frac{31}{0.10} = 310\) reveals the initial price.
This solution demonstrates how to translate a real-world problem into a mathematical equation and solve it step by step. Once you become skilled at such techniques, you can tackle a wide array of problems with confidence.