Question

You are budgeting money to buy a violin and bow that cost \(\$ 250\) including tax. If you save \(\$ 5\) per week, will you have enough money in a year? You write the inequality \(5 n \geq 250\) to model the situation. What do \(5, n,\) and 250 represent? Solve the inequality.

Short answer

Yes, saving $5 per week for one year would be enough to buy the violin and bow. In the inequality \(5n \geq 250\), \(n\) represents the number of weeks, \(5\) represents the weekly savings, and 250 is the cost of the violin and bow. Solving the inequality shows that at least 50 weeks are necessary to have enough savings, which is less than a year (52 weeks).

Step-by-step solution

Identify the Variables

In the given inequality \(5n \geq 250\), \(5\) represents the amount saved per week, \(n\) represents the number of weeks, and 250 represents the total cost of the violin and bow, all in dollars.

Solve the Inequality

To solve for \(n\), we need to isolate \(n\) on one side of the inequality. To do this, we will divide each side of the given inequality by $5. We get \(n \geq 250 \div 5\), which reduces to \(n \geq 50\). This inequality tells us the number of weeks needed to have enough savings to buy the violin and bow.

Interpret the Results

Since a year has 52 weeks and our calculation shows we need at least 50 weeks, we can conclude that saving $5 each week for a year will indeed be enough to purchase the violin.

Key concepts

Inequalities in Algebra

Inequalities are like the siblings of equations in algebra; they show the relationship between expressions that are not necessarily equal but may have a greater-than or less-than relationship. An understanding of inequalities is crucial in both math and real-life situations, such as budgeting finances or comparing quantities.

In the context of the exercise, an inequality is used to determine if saving \(\$5\) per week will lead up to the necessary amount to buy an item—the violin and bow costing \(\$250\). The inequality \(5n \geq 250\) utilizes algebraic expressions and variables to represent the situation. Here, \(5\) represents the weekly saving amount, \(n\) is the number of weeks, and \(250\) is the total cost that needs to be met or exceeded. By manipulating the inequality, it's possible to find a solution that indicates whether the goal can be achieved within a specific timeframe such as a year. This real-life application demonstrates the importance of understanding and solving inequalities in algebra.

Budgeting and Saving

Grasping the concepts of budgeting and saving is essential for making sound financial decisions. In our exercise, the simple act of saving \(\$5\) each week translates into an algebraic model, reminding us that financial planning can often be represented mathematically.

Budgeting requires setting financial targets, like the violin and bow purchase, and then working out how to achieve them. By saving consistently over time (\(n\) weeks), these targets can be met. The inequality \(5n \geq 250\) shows that cumulative savings need to reach or surpass a specific benchmark, in this case, \(\$250\). The solution to the inequality provides not just an answer, but also a strategic insight: if we save for at least 50 weeks, we will have enough to meet our goal, highlighting the practical use of algebra in personal finance strategy.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols that represents a specific value or set of values. For someone learning algebra, understanding these expressions is as fundamental as understanding sentences in a language.

In our problem, the expression \(5n\) encapsulates the saving plan—a multiplying factor of 5 (the money saved each week) and the variable \(n\) (the number of weeks). This type of expression allows us to model real-world scenarios algebraically and solve for unknowns. Breaking down and interpreting these expressions is a key algebra skill, which is applied here to find out how long it'll take to save up enough money to achieve a budgeting goal. Expressions provide a succinct and powerful way to handle diverse problems, often making complex scenarios much simpler to work through.