Problem 31 - Prealgebra

Question

Antonia bought a video game system for \(\$ 323.96 .\) She paid in 12 equal installments. Which is the best estimate for the amount of each payment? A. less than \(\$ 20\) B. between \(\$ 20\) and \(\$ 25\) C. between \(\$ 25\) and \(\$ 30\) D. greater than \(\$ 30\)

Step-by-Step Solution

Answer

C: between $25 and $30.

Step 1: Understanding the Problem

Antonia bought a video game system for a total amount of $323.96 and paid for it over 12 equal monthly installments. We are asked to estimate the amount of each monthly payment.

Step 2: Setup the Division

To find the amount of each installment, we need to divide the total cost of the video game system, $323.96, by the number of installments, which is 12.

Step 3: Calculation

Perform the division to find the amount of each installment: \[ 323.96 \div 12 \approx 27 \]

Step 4: Matching with Estimates

Using the calculation, the result of 27 falls between $25 and $30. Check against the given options: A (less than $20), B (between $20 and $25), C (between $25 and $30), D (greater than $30). The best estimate is option C.

Key Concepts

Division

Division is a fundamental mathematical operation where we distribute a number into equal parts. In the context of financial payments, it helps us understand how a total cost can be broken down into smaller, manageable portions. For example, when Antonia wants to pay off her video game system, which costs \(323.96, in equal monthly installments, we need to divide the total amount by the number of months. This is why we use division. It gives us a clear view of how much will be paid each month.

To perform this division, we calculate \(323.96 \div 12\), resulting in approximately \)27 per month. Division thus becomes powerful, enabling us to take complex, larger values and simplify them into smaller, more practical amounts. This is essential in both everyday life and business, where payments often need precise division over time frames.

Monthly Installments

Monthly installments are a common approach to managing payments for both large and small purchases. When you buy something expensive, like a video game system, paying the whole amount upfront can be difficult. Splitting the total cost into monthly payments makes it affordable and easier to manage within a monthly budget.

In Antonia's situation, paying $323.96 at once might be a stretch. However, breaking it down into 12 installments transforms this cost into smaller monthly payments. This allows for better financial planning and budgeting, ensuring that payments are met without financial stress. Thus, understanding how monthly installments work can greatly enhance personal financial management.

Financial Literacy

Financial literacy involves understanding various aspects of personal and business finance, including the ability to analyze and apply different payments and financial scenarios. Having a good grasp of financial concepts such as division and monthly installments is crucial to making informed decisions.

For instance, Antonia, through dividing the total cost into installments, demonstrates financial literacy by efficiently managing her resources. She is taking a proactive approach to her finances rather than spending beyond her means. Improving financial literacy equips individuals with the knowledge to make smarter money decisions, save more efficiently, and avoid unnecessary debt. It's a critical life skill that can impact one's financial health and stability.

Problem-Solving

Problem-solving in mathematics involves breaking down a problem into manageable steps and finding a logical solution. The task of estimating monthly payments for a product is a great example of applying problem-solving skills in real life.

Let's look at how this is applied in Antonia's scenario:

• First, she understands the total cost and the number of installments.
• She then sets up the division of total cost by the number of installments to find an approximate installment amount.
• After calculating, she matches this estimate with predefined options, demonstrating logical assessment.

By following these steps, Antonia not only arrives at an accurate estimate but also enhances her analytical and mathematical skills through practice and application. Effective problem-solving showcases the capacity to think critically and arrive at sensible conclusions, a valuable skill in both academic and everyday situations.