Bryan's Six-Step Solution To A Mathematical Problem

Bryan employed a structured six-step approach to tackle a mathematical problem involving the cost of boxes. This method, widely recognized in problem-solving literature, encourages a systematic and logical progression from understanding the problem to verifying the solution. Let's delve into Bryan's journey, exploring each step and how it contributed to his final answer.

Understanding the Problem: Decoding the Cost of Boxes

In the crucial first step, understanding the problem, Bryan meticulously analyzed the given information. This stage is paramount as it lays the foundation for the entire solution process. The problem presented two options for purchasing boxes, each with a distinct pricing structure:

Option A offered boxes at a fixed rate of $14.65 per box. This seemed straightforward – the total cost would be directly proportional to the number of boxes purchased. For example, buying two boxes would cost $29.30 (2 x $14.65), and so on.

Option B presented a slightly more complex scenario. The first box was offered free of charge, a tempting proposition. However, any additional boxes beyond the first would cost $16.20 each. This introduces a variable cost element, where the total expense depends on the number of boxes purchased after the initial free one.

Bryan's task at this stage was not just to passively read the information but to actively engage with it. He likely asked himself clarifying questions such as:

• What exactly is being asked? Is it to determine the cost for a specific number of boxes under each option? Or perhaps to find the number of boxes for which the costs are equal?
• What are the key pieces of information? Clearly, the prices per box ($14.65 and $16.20), and the “first box free” condition are critical.
• Are there any hidden assumptions or constraints? For instance, is there a limit to the number of boxes that can be purchased?

By thoroughly understanding the problem, Bryan set himself up for success. He had a clear mental picture of the situation and what needed to be solved. This initial investment of time and effort would prove invaluable in the subsequent steps.

Devising a Plan: Charting the Course to a Solution

With a firm grasp of the problem, Bryan moved on to the second step: devising a plan. This involves strategizing how to bridge the gap between the given information and the desired solution. Effective planning is the cornerstone of efficient problem-solving. Bryan likely considered several approaches, weighing their pros and cons before settling on a course of action.

One possible strategy could have been to create a table or a chart. This visual representation would allow him to compare the cost of different quantities of boxes under each option. He could list the number of boxes (1, 2, 3, and so on) in one column, the cost under Option A in another, and the cost under Option B in a third. This tabular format would make it easy to identify patterns and pinpoint the point at which one option becomes more cost-effective than the other.

Another approach could involve using algebraic equations. Bryan could represent the number of boxes as a variable (say, 'x') and formulate equations for the total cost under each option. For Option A, the cost would be 14.65x. For Option B, it would be 16.20(x-1) since the first box is free. By setting these equations equal to each other, Bryan could solve for 'x', which would represent the number of boxes at which the costs are the same.

Yet another possibility is to use a graphical method. Bryan could plot the cost of each option as a function of the number of boxes. The point where the two lines intersect would represent the break-even point, where the costs are equal. This visual approach can provide an intuitive understanding of the problem.

Bryan's choice of plan would likely depend on his individual preferences and strengths. The key is that he consciously chose a method that he believed would lead him to the solution in a systematic and organized manner. This step highlights the importance of strategic thinking in problem-solving. Rather than blindly jumping into calculations, Bryan took the time to devise a roadmap, ensuring that his efforts would be focused and productive.

Carrying Out the Plan: Executing the Chosen Strategy

Having formulated a plan, Bryan proceeded to the third step: carrying out the plan. This is where the chosen strategy is put into action. Careful execution is crucial at this stage, as any errors in calculation or logic can derail the entire process. Let's imagine Bryan chose to use the algebraic equation method, as it offers a precise and efficient way to solve the problem. His steps might look something like this:

1. Define variables: Let 'x' represent the number of boxes.
2. Formulate equations:
   • Cost under Option A: 14.65x
   • Cost under Option B: 16.20(x-1) (since the first box is free)
3. Set the equations equal: To find the number of boxes where the costs are the same, Bryan would set the two equations equal to each other: 14.65x = 16.20(x-1)
4. Solve for x: This involves algebraic manipulation:
   • 14. 65x = 16.20x - 16.20
   • 15. 20 = 1.55x
   • x = 16.20 / 1.55
   • x ≈ 10.45

Bryan would perform these calculations meticulously, paying close attention to detail. He might use a calculator to ensure accuracy and double-check his work at each step. The goal is to execute the plan flawlessly, minimizing the risk of errors.

If Bryan had chosen a different approach, such as the tabular method, he would systematically calculate the cost for each number of boxes under both options and record them in a table. He would then compare the costs to identify the break-even point.

The key takeaway is that this step is about methodical execution. Bryan is putting his plan into action, step by step, until he arrives at a potential solution. This requires focus, precision, and a commitment to following the chosen strategy through to its conclusion.

Looking Back: Verifying and Interpreting the Solution

With a potential solution in hand, Bryan arrived at the fourth step, a critical stage often overlooked: looking back. This involves verifying the solution and interpreting its meaning in the context of the original problem. It's not enough to simply arrive at a numerical answer; Bryan needed to ensure that the answer made sense and that it actually addressed the question asked.

In Bryan's case, the algebraic solution yielded x ≈ 10.45. This suggests that the costs of the two options are equal when approximately 10.45 boxes are purchased. However, boxes are typically sold in whole units, so Bryan needed to interpret this result carefully. He couldn't buy a fraction of a box. Therefore, he needed to consider the costs for 10 boxes and 11 boxes under each option.

He might create a small table to compare these costs:

This table reveals that Option B is slightly cheaper for 10 boxes, while Option A becomes cheaper for 11 boxes or more. This detailed analysis allows Bryan to not only verify the numerical result but also to gain a deeper understanding of the problem's dynamics.

Looking back also involves checking for errors in the calculations or reasoning. Bryan might retrace his steps, reviewing each stage of the solution process to ensure accuracy. He might also consider whether the answer is reasonable in the context of the problem. For instance, if he had arrived at a negative number of boxes, he would immediately recognize that something went wrong.

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