Problem 59 - Algebra 1

Question

Use the following information. You are shopping for a mountain bike. A store sells two different models. The model that has steel wheel rims costs \(\$ 220 .\) The model with aluminum wheel rims costs \(\$ 480 .\) You have a summer job for 12 weeks. You save \(\$ 20\) per week, which would allow you to buy the model with the steel wheel rims. You want to know how much more money you would have to save each week to be able to buy the model with the aluminum wheel rims. Write a verbal model and an algebraic model for how much more money you would have to save each week.

Step-by-Step Solution

Answer

To purchase the aluminum-wheel-rimmed model, an extra \$10 per week is required, bringing the total weekly savings to \$30.

Step 1: Identify Known Values

The known values are: the cost of the steel-rimmed model (\$220), the cost of the aluminum-rimmed model (\$480), the savings per week for the steel-rimmed model (\$20), and the duration of the summer job (12 weeks).

Step 2: Construct a Verbal Model

You may start by constructing a verbal model. For the steel-rimmed model, saving \$20 per week for 12 weeks gives a total savings of \$240. However, if the objective is to buy the more expensive aluminum-rim model, the saved money (\$240) falls short of the required \$480. Therefore, a certain amount of extra savings per week is needed to achieve the goal.

Step 3: Construct an Algebraic Model

The algebraic model will depict the relationship mathematically. Let x reflect the increased savings needed. The current total savings (20*12) added with the increase (\$ x) for 12 weeks should reach the cost of the more costly bike. The equation can be formulated as follows: 20 * 12 + x * 12 = 480.

Step 4: Solve the Algebraic Model

The algebraic model can be solved by isolating the variable x: x = (480 - 20*12) / 12.

Key Concepts

Verbal Model

When working with algebraic problems, creating a verbal model is the first step to reaching a solution. A verbal model helps you translate a problem described in words into a form you can work with mathematically. In this exercise, the problem is buying a specific model of mountain bike. Initially, you save \(20 per week for the 12-week duration of your summer job, giving you a total savings of \(12 \times 20 = 240\). However, to buy the more expensive aluminum-rimmed bike costing \\)480, you need additional savings.

Here's how you create a verbal model for the problem:

• Total savings with current plan: \(240
• Cost of the aluminum-rimmed bike: \)480
• Extra savings needed each week to afford the more costly bike

By breaking down each part of the problem, you can visualize what's needed to achieve your goal.

Algebraic Model

Creating an algebraic model allows you to express the verbal model in mathematical terms with variables and equations. This step is crucial since it provides the means to calculate exactly what you need to do to achieve the desired outcome.

• Let the variable \(x\) represent the extra amount you ought to save each week.
• The equation representing the total needed savings involves multiplying this extra savings by 12 weeks, matching the timeline of your summer job.

Your total equation to determine the needed savings could be expressed as:\[20 \times 12 + x \times 12 = 480\]This equation helps to mathematically represent the gap between current savings and the cost of the aluminum-rimmed bike.

Equation Solving

Solving the algebraic model is the next pivotal step. It involves manipulating the equation to find the unknown quantity, providing a precise answer to the problem.

• First, calculate the total current savings: \(20 \times 12 = 240\).
• Rearrange the equation: \(240 + 12x = 480\).
• Solve for \(x\) by isolating it on one side of the equation:

Start by subtracting 240 from both sides: \(12x = 480 - 240\). Continue by reducing this to \(12x = 240\). Finally, divide both sides by 12 to solve for \(x\):\[x = \frac{240}{12} = 20\]This result tells you that saving an additional $20 per week will enable the purchase of the aluminum-rimmed bike by the end of the 12-week period.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to see how each step in solving a problem connects to reach the final goal. It helps ensure that both the approach and solution are sound.

• By starting with a verbal model, you clearly outline the problem's parameters and goals.
• Transforming this into an algebraic model, you employ mathematical symbols to succinctly express relationships between different quantities.
• Solving the equation confirms your intuition and supports it with concrete numbers.

Mathematical reasoning doesn't merely aim at completing calculations. It's about understanding why each step matters and confirming the feasibility of solutions in real-life contexts, such as budget planning and purchasing within savings limits.