Problem 18 - Algebra 1: Concepts and Skills

Question

An appliance store sells two stereo models. The model without a CD player is \(\$ 350 .\) The model with a CD player is \(\$ 480 .\) Your summer job allows you to save \(\$ 50\) a week for 8 weeks. At the end of the summer, you have enough to buy the stereo without the CD player. How much would you have needed to save each week to buy the other model? Use the labels to translate your verbal model into an equation.

Step-by-Step Solution

Answer

The weekly savings required to buy the stereo with the CD player is \$60.

Step 1: Translate the Problem into an Equation

Identify the known constants in the problem: the cost of the two stereos (\$350 for the model without the CD player and \$480 for the model with the CD player) and the length of the summer job (8 weeks). The weekly savings (\$50) allowed to purchase the cheaper model. To find out the weekly savings to buy the more expensive model, an equation can be set up as: Savings_per_week x total_weeks = cost_of_expensive_model

Step 2: Substitute the Known Constants

Substitute the known constants into the equation from Step 1: Savings_per_week x 8 = \$480.

Step 3: Solve the Equation

Solve the above equation for Savings_per_week. This can be done by dividing both sides of the equation by 8. This leads to Savings_per_week = \$480 / 8.

Step 4: Calculate the Value

Now perform the division operation on the right side of the equation. This gives, Savings_per_week = \$60.

Key Concepts

Linear Equations in Real-Life Situations

Linear equations are a powerful tool in solving real-life problems, like budgeting for purchases. These equations have the standard form: \(ax + b = c\), where \(x\) represents an unknown that you need to find. In this exercise, our goal is to discover how much should be saved weekly to afford the costlier stereo model.

By setting up a linear equation, we used the formula: \(\text{Savings\_per\_week} \times \text{total\_weeks} = \text{cost\_of\_expensive\_model}\). Here, \(\text{Savings\_per\_week}\) is the variable we're solving for. These real-world math scenarios help us make informed financial decisions.

Budgeting and Saving Calculations

Effectively managing finances involves understanding budgeting and saving. In this situation, the goal was a more expensive stereo, costing \\(480. We needed to determine the amount to save each week over 8 weeks.

Calculating this requires setting a savings target that matches the cost of your desired purchase. Initially, saving \\)50 a week was enough for the cheaper model but not the pricier one. Adjusting the budget to \$60 per week helped meet the goal.

• Determine the total amount needed for your goal.
• Divide this total by the number of weeks or months available.
• Adjust your weekly savings plan as necessary to achieve your goal.

These are key steps in planning a budget.

Mastering Arithmetic Operations

Arithmetic operations, like multiplication and division, are fundamental in solving equations. In our scenario, they were used to compute the savings needed weekly. Starting with the equation \(8 \times \text{Savings\_per\_week} = 480\), dividing both sides by 8 simplified the problem.

This operation resulted in \(\text{Savings\_per\_week} = 60\).

• Understand each operation's purpose: multiplication for scaling and division for finding individual portions.
• Use basic mathematical rules to isolate variables.
• Simplify complex problems into smaller, manageable steps.

These skills are crucial not just in academics but also in daily financial decision-making.