Problem 17 - 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? Assign labels to your verbal model. Use \(m\) to represent the unknown value.

Step-by-Step Solution

Answer

You would need to save \$60 each week to be able to afford the stereo with the CD player.

Step 1: Establish Amount Earned Over Summer

Firstly,multiply the amount saved each week with the number of weeks. Thus, for the model without a CD player: \(50 \times 8 = 400 \) dollars.

Step 2: Determine Price Difference between Models

Secondly, calculate the difference in price between the two models to determine how much more needed to be saved. This is done by subtracting the price of the model without a CD player from the price of the model with a CD player: \(480 - 350 = 130 \) dollars.

Step 3: Determine Weekly Saving for the Expensive Model

Lastly, divide the total amount needed for the model with a CD player (480 dollars) by the number of weeks (8). Thus, \(m = \frac{480}{8} = 60 \) dollars per week (where m represents the unknown value).

Key Concepts

Linear Equations

Linear equations are essential tools in mathematics and are used to model relationships between variables. In the context of algebra word problems, such equations help us calculate unknown values based on given information. A linear equation typically has the structure of \( ax + b = c \), where \( x \) represents the unknown value we aim to determine.

In the problem of purchasing a stereo with weekly savings, we use a linear equation to establish how much we need to save each week. For example, if we know the total cost of the stereo and the number of weeks available to save, we can express the amount saved each week as \( m = \frac{480}{8} \), where \( m \) denotes the weekly savings required.

This equation helps you see the direct relationship between weekly savings, the number of weeks, and the total amount needed for the purchase.

Price Difference

Understanding price difference is crucial when comparing the costs of different items. It's straightforward: we subtract the lower price from the higher price to determine the difference.

In this exercise, we calculate the price difference between the stereo models by subtracting the cost without the CD player from the cost with it: \( 480 - 350 = 130 \) dollars.

The significance of knowing the price difference lies in budgeting and planning savings. It helps identify how much extra money is required to upgrade or purchase a more expensive alternative. In this case, knowing that the other model costs an additional \$130 allows you to plan your savings accordingly for a smarter financial decision.

Weekly Savings Calculation

Calculating weekly savings involves determining how much money needs to be set aside each week to achieve a specific financial goal over a set period.

In the given problem, you save \\(50 each week for 8 weeks to accumulate enough for a \\)350 stereo. Through calculation, \( 50 \times 8 = 400 \) dollars is saved, showing efficiency compared to the stereo price.

It's crucial to adjust weekly savings when goals change, like buying a \$480 stereo. The new plan involves a higher weekly saving rate: \( m = \frac{480}{8} = 60 \) dollars. These insights ensure financial goals are met within timelines, making budgeting and saving practical and achievable for future purchases.