Problem 58 - Algebra 1: Concepts and Skills

Question

BUYING A GUITAR You are budgeting money to buy a guitar that 150 dollars including tax. If you save 20 dollars per month, will you have enough money in 6 months? Use the inequality \(20 n \geq 150\) to model the situation, where \(n\) represents the number of months.

Step-by-Step Solution

Answer

With $20 saving per month, you will have $120 in six months, which is not enough to buy a guitar costing $150. Therefore, it will not be possible to buy the guitar in six months with this saving plan.

Step 1: STEP 1: UNDERSTAND THE INEQUALITY

The inequality \(20n \geq 150\) means that the money saved from 'n' months (which is $20 per month) needs to be greater than or equal to $150, which is the cost of the guitar.

Step 2: STEP 2: SUBSTITUTION

Replace \(n\) in the inequality with 6 months, which changes the inequality into \(20*6 \geq 150\).

Step 3: STEP 3: SOLVING THE INEQUALITY

Solve the inequality \(20*6 \geq 150\). This simplifies to \(120 \geq 150\).

Step 4: STEP 4: EVALUATION

Analyze whether the inequality \(120 \geq 150\) holds true. In this case, 120 is not greater than or equal to 150, so the inequality is not true.

Key Concepts

Budgeting

Managing money effectively is essential for reaching financial goals, such as buying something you desire, like a guitar. Budgeting is the process of creating a plan to spend your money wisely. It involves estimating your income and expenses over a certain period, which helps you understand how much you can save and spend. For students, this can be especially useful to manage allowances or part-time job income. Having a budget lets you make informed decisions and ensures you have enough money set aside for important purchases.
Using our example, we understand the monthly income (savings) is $20. The goal is to buy a guitar worth $150. By drafting a budget, you identify how you need to adjust your spending and saving habits to achieve this goal. Budgeting might also highlight the need to increase savings, perhaps by cutting down on unnecessary expenses to make the goal achievable.

Saving Money

Saving money is a fundamental component of reaching financial goals. It is setting aside a portion of your income regularly to help you prepare for future expenses or purchases. In our example, saving $20 each month reflects a consistent approach aimed at gathering enough funds to buy a guitar. When you save regularly, it ensures you accumulate sufficient funds over time.
While $20 a month might seem like a small amount, consistent savings can grow exponentially. It's always a good practice to reassess your saving strategies and consider:

• Setting clear savings goals
• Monitoring your progress regularly
• Finding additional ways to save money, such as earning extra income

In our scenario, reaching the required $150 in 6 months was not feasible with $20 monthly savings. Therefore, looking into alternative saving methods or saving for a larger number of months would be necessary.

Algebraic Modeling

Algebraic modeling is a crucial skill that allows you to represent real-world situations with mathematical expressions. It simplifies complex problems, making them easier to analyze and solve. By using variables and operations, algebra helps you to predict and interpret outcomes.
In the guitar-buying scenario, the inequality model used was \(20n \geq 150\), where \(n\) represents the number of months. This model helps visualize if your monthly savings cover the cost of the guitar within a set timeline.
Analyzing this through substitution (\(n = 6\)), we found that \(120 \geq 150\) is false, indicating the savings would not cover the total cost in 6 months. Algebraic modeling hence provides a clear picture of what adjustments are necessary, such as increasing the number of saving months or increasing the monthly saving amount. By learning to model situations algebraically, students can tackle everyday financial decisions with mathematical reasoning.