Question

Caitlin opens a checking account that earns no interest to set aside spending money for vacations. Each month she puts the same dollar amount, \(\$ 50\), into the account. Unfortunately, she does not expect to be able to take a vacation at any point in the foreseeable future. Which of the following best describes the relationship between the number of months and the total amount of money in the account? (A) A linear relationship, with the line of the relationship having a negative slope B) A linear relationship, with the line of the relationship having a positive slope (C) An exponentially increasing relationship (D) An inverse exponential relationship

Short answer

Option B: A linear relationship with a positive slope.

Step-by-step solution

Understand the Problem

Caitlin is depositing a fixed amount of $50 each month into a checking account. We want to determine how the total amount in the account changes over time and describe this relationship.

Define the Variables

Let \( x \) represent the number of months Caitlin deposits money into her account, and \( y \) represent the total amount of money in the account.

Establish the Relationship

Each month, Caitlin deposits an additional \( \$50 \), leading to a direct addition to her account. This creates a linear relationship between \( x \) and \( y \), defined by \( y = 50x \), where \( y \) is the total amount in the account after \( x \) months.

Determine the Slope of the Line

The equation \( y = 50x \) represents a linear function with a slope of 50. As the number of months (\( x \)) increases, the total amount of money (\( y \)) increases by 50 for each additional month, indicating a positive slope.

Compare with Answer Choices

We identified that this is a linear relationship with a positive slope. Comparing this with the given options, the correct choice matches option B, "A linear relationship, with the line of the relationship having a positive slope."

Key concepts

Mathematical Functions

Mathematical functions are essential tools in describing various relationships. In Caitlin's scenario, the relationship between the number of months and the money saved can be expressed using a linear function. A linear function is one in which a constant rate of change exists. This means for each unit increase in one variable, the other variable changes by a fixed amount.

In this case, this consistent change is captured by the equation \( y = 50x \). Here \( y \), the total amount of money, changes linearly with respect to \( x \), the number of months. The constant 50 is the slope, illustrating that for every month Caitlin saves, her account increases by $50.

Understanding mathematical functions allows us to predict future outcomes and identify patterns. With a linear function, you always end up forming a straight line when you graph the relationship. This means there is a balance, as every input corresponds to a single, predictable output.

Financial Literacy

Financial literacy encompasses knowledge and skills that allow individuals to make informed and effective financial decisions. One practical application is understanding savings vehicles, like Caitlin’s checking account scenario.

Opening a checking account for savings can be advantageous, especially for regular deposits. Caitlin’s consistent monthly deposits represent a simple savings strategy that builds wealth gradually. Though the account earns no interest, the discipline of regular savings accrues a sizable amount over time.
- Keeps funds secure for future use - Encourages consistent saving habits

The decision of where and how to save is vital in financial literacy. While Caitlin's account does not accrue interest, understanding mathematical growth, like linear increases, empowers her to visualize and quantify her savings progress over time. By recognizing the potential in even basic saving strategies, Caitlin builds towards her financial goals effectively.

Problem-Solving Skills

Problem-solving skills involve the ability to understand a problem, devise a solution, and adjust the approach as needed. Let's break down Caitlin's example as an exercise in problem-solving.

First, identify the problem: Caitlin wants to determine how her account balance grows over time. Translating this into a mathematical function is the second step. By defining variables, like months and total savings, and establishing the equation \( y = 50x \), you describe the saving process linguistically and mathematically.
- Identify the problem and clarify the goal.- Define variables and establish relationships.
Finally, evaluating and checking the solution involves interpreting the linear equation and confirming that its description matches the expected outcome. This crucial step ensures Caitlin's understanding matches reality, thereby honing both analytical and practical problem-solving skills.A thorough understanding of how to approach problems like Caitlin's enhances critical thinking, encouraging more effective financial management and planning.