Problem 60 - 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. Use mental math to solve the equation. What does the solution represent?

Step-by-Step Solution

Answer

The student has to save an additional \$20 per week to buy the model with the aluminum wheel rims.

Step 1: Calculate Total Savings

First, let's calculate the total savings from the summer job. The student has a job for 12 weeks and saves \$20 each week. This can be calculated as \(12 \times 20 = \$240\). So, the total savings after 12 weeks is \$240.

Step 2: Calculate the Additional Required Amount

Next, the cost of the aluminum wheel rim model is \$480. The total savings currently are \$240. Therefore, the additional amount needed can be calculated as \(\$480 - \$240 = \$240\). So, the student needs to save an additional \$240 to buy the aluminum rim model.

Step 3: Find Out the Weekly Saving Needed

The student needs to save an additional amount of \$240 over 12 weeks. Divide the total amount by the number of weeks to calculate the extra saving required per week. Hence, the extra saving needed each week can be found as \(\$240 ÷ 12 = \$20\). This means the student must save \$20 more per week.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a cornerstone of mathematics, serving as a toolkit for unraveling real-world situations and converting them into equations to find solutions. It involves understanding the problem context, identifying the variables involved, and formulating a mathematical representation that can be systematically solved.

In the context of our exercise, algebraic problem solving begins by identifying our ultimate goal: determining the additional weekly savings needed to purchase the aluminum wheel rim model. Equipped with the price difference between the two bike models and the duration of the savings period (12 weeks), one can set up an equation to represent the situation. Here, the total additional amount (\$240) divided by the number of weeks (12) gives us the extra amount to save per week (\$240 / 12 = \(20\)). Algebraic problem solving requires an analytical approach to break down the problem into smaller, manageable parts, and solve it step by step.

Linear Equations

Linear equations form the backbone of many mathematical applications and describe a straight-line relationship between two variables. They can often be written in the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

In our bike model example, while it may not seem obvious at first, we are essentially dealing with a linear relationship. The total savings can be considered a linear function of the amount saved each week. The algebraic setup, based on the given problem, did not require the explicit formulation of a line equation, but understanding the concept of linearity helps to see that saving more per week (increasing the value of \( m \)) leads to a proportional increase in total savings. Using mental math to solve linear equations encourages students to internalize the principles of linearity and develop a deeper understanding of how different variables affect each other.

Mathematical Reasoning

Mathematical reasoning involves logical thinking, problem-solving, and the ability to draw appropriate conclusions based on mathematical principles. It challenges students to analyze information, make connections, and justify their thought process.

Within the solution to our bike savings problem, mathematical reasoning is demonstrated by interpreting the equation's solution: the additional \(20\) dollars per week represents the increase in weekly savings necessary to afford the more expensive bike model. The student's logical deduction that doubling the original weekly savings will allow them to reach their goal is a great illustration of mathematical reasoning in action. This level of thinking enables students to apply mathematical concepts to similar problems and different contexts, promoting a transferable, problem-solving mindset.