Calculating Gift Card Balance After App Purchases A Math Example

Let's delve into a scenario involving Hillary and her gift card, focusing on how the balance changes as she purchases apps. This problem combines mathematical concepts with a practical, real-world situation, making it both engaging and relatable. Our goal is to understand the equation that represents the remaining balance on Hillary's gift card and use it to calculate her balance after a specific number of app purchases. We'll also explore the broader implications of such linear equations in managing finances and understanding spending habits.

Setting the Stage: Hillary's Gift Card

Imagine Hillary receives a gift card with a starting balance of $50. She decides to use this gift card to purchase apps for her phone. Each app costs $1.25. We want to create a mathematical model that represents the amount of money remaining on her gift card after she purchases a certain number of apps. This model will allow us to predict her remaining balance after any number of purchases, providing a clear picture of her spending and the depletion of her gift card funds.

The core concept here is understanding how a fixed amount (the initial gift card balance) decreases with each purchase. This decrease is consistent, as each app has the same cost. This scenario perfectly lends itself to a linear equation, where the amount spent increases linearly with the number of apps purchased. Let's break down the components of this equation to see how it accurately models Hillary's spending.

The Equation: a(x) = 50 - 1.25x

The equation provided, a(x) = 50 - 1.25x, is a linear equation that models the amount of money remaining on Hillary's gift card. Let's dissect this equation to understand its components:

• a(x): This represents the amount of money remaining on the gift card after purchasing x apps. It's the dependent variable, meaning its value depends on the value of x.
• 50: This is the initial amount of money on the gift card, the starting point. It's the y-intercept in the linear equation, representing the value of a(x) when x is zero (no apps purchased).
• 1.25: This is the cost of each app. It's the rate at which the gift card balance decreases with each app purchase. This is the slope of the linear equation, indicating the change in a(x) for each unit increase in x.
• x: This represents the number of apps Hillary purchases. It's the independent variable, the value we can change to see its effect on the remaining balance.

The equation works by subtracting the total cost of the apps (1.25 multiplied by the number of apps, x) from the initial balance of $50. This accurately reflects how Hillary's gift card balance decreases with each purchase. Now, let's put this equation to work by calculating Hillary's balance after she buys 12 apps.

Calculating the Remaining Balance: a(12)

The question asks us to determine a(12), which means we need to find the amount of money remaining on Hillary's gift card after she purchases 12 apps. To do this, we simply substitute x with 12 in the equation:

• a(12) = 50 - 1.25 * 12

First, we perform the multiplication:

• 1. 25 * 12 = 15

Then, we subtract this result from the initial balance:

• a(12) = 50 - 15
• a(12) = 35

Therefore, a(12) = 35. This means that after Hillary purchases 12 apps, she will have $35 remaining on her gift card. This calculation demonstrates the power of the equation in predicting the remaining balance after any number of app purchases. We can use this same method to calculate the balance after 5 apps, 20 apps, or any other number.

Implications and Further Exploration

This simple scenario highlights the practical application of linear equations in managing personal finances. By understanding the relationship between spending and remaining balance, we can make informed decisions about our purchases. For example, Hillary can use this equation to determine how many more apps she can purchase before her gift card balance reaches zero.

Furthermore, this concept can be extended to various other financial situations, such as budgeting, loan repayment, and investment growth. Linear equations provide a foundational understanding of how quantities change over time, making them a valuable tool in financial planning.

In conclusion, the equation a(x) = 50 - 1.25x accurately models the amount of money remaining on Hillary's gift card after purchasing x apps. By substituting x with 12, we found that a(12) = 35, meaning Hillary has $35 left on her gift card after buying 12 apps. This exercise demonstrates the practical application of linear equations in everyday scenarios and their importance in financial literacy.

The original input keyword essentially asks for the evaluation of the function a(x) at x = 12, given the function a(x) = 50 - 1.25x. To make the question even clearer and more direct, we can rephrase it as follows:

"Given the function a(x) = 50 - 1.25x, what is the value of a(12)?"

This rephrased question is concise and immediately focuses on the core task: evaluating the function at a specific point. It removes any ambiguity and ensures that the user understands the objective without needing to interpret the context or scenario.

Here's a breakdown of why this rephrased question is an improvement:

• Directness: The question directly asks for the value of a(12) without any introductory context.
• Clarity: It explicitly states the function a(x) and the value at which it needs to be evaluated.
• Conciseness: The question is shorter and more to the point, making it easier to understand.
• Focus: It immediately highlights the mathematical operation required (function evaluation).

By rephrasing the question in this way, we ensure that the user understands the task at hand and can focus on applying the appropriate mathematical steps to find the solution. This is particularly important in educational settings where clarity and precision are paramount.