Question

Your department sends its copying to a photocopy center. The photocopy center bills your department \(\$ 0.08\) per page. You are considering buying a departmental copier for \(\$ 2500 .\) With your own copier the cost per page would be \(\$ 0.025 .\) The expected life of the copier is 4 years. How many copies must you make in the four-year period to justify purchasing the copier?

Short answer

You need to make at least 45455 copies in the four-year period to justify purchasing the copier.

Step-by-step solution

Determine the costs

Firstly, the costs of both the options are known. For the photocopy center, the cost per page is \(0.08\). For the own copier, the cost is the cost per page (\(0.025\)) plus the cost of the copier itself (\(2500\)).

Formulate the equation

Next, the total cost for each option can be formulated. Let 'x' be the number of copies. So, for the photocopy center, the total cost will be \(0.08x\). For the departmental copier, the total cost will be \(2500 + 0.025x\). To find the breakeven point, these two costs should be equated.

Solve the equation

Hence the equation is \(0.08x = 2500 + 0.025x\). Solving this for 'x' will give us the breakeven point in terms of copies.

Calculate the breakeven point

Solving equation gives \(0.055x = 2500\). So, \(x = 2500 / 0.055 = 45454.54\). Since we can't have a fraction of a copy, the number of copies is rounded up to the next whole number, i.e., 45455 copies

Key concepts

Cost-Benefit Analysis

The process of weighing the costs against the benefits of any decision or investment is known as cost-benefit analysis. This is a fundamental concept in finance and economics that helps determine whether a particular course of action is financially viable or not. In the context of the provided exercise, the cost-benefit analysis involves comparing the costs of using an external photocopy center versus the costs of purchasing and operating a departmental copier.

When conducting a cost-benefit analysis for buying equipment like a copier, it's crucial to consider both the initial capital expense and the ongoing operational costs. It's also necessary to project the costs for an equivalent time period for both options to make a fair comparison. In the exercise, the lifetime of the copier is given as four years, therefore the benefits, or savings, must also be calculated over the same time span.

Understanding the break-even point is important in cost-benefit analysis as it tells us the minimum performance level required (in this case, the number of copies) to justify a decision (purchasing the copier). After reaching the break-even point, any additional copies made result in net savings for the department.

Algebraic Equations

Algebraic equations are mathematical statements that declare the equality of two expressions, often involving a variable which needs to be solved. They are an essential tool in problem-solving across various disciplines, from finance to physics. In the photocopy example, the algebraic equation is used to find the number of copies that need to be made for the cost of purchasing a copier to equal the cost of outsourcing the copying. The core of algebra involves manipulating these equations to solve for the unknown value.

In setting up the equation, the challenge is to accurately represent real-world situations with mathematical expressions. The equation \(0.08x = 2500 + 0.025x\) mirrors the cost structure of both copy options, incorporating the per-page cost and the initial investment. The equation is then rearranged to isolate the variable, thereby facilitating the discovery of the exact number of copies that represent the break-even point.

Problem-Solving in Mathematics

Problem-solving in mathematics is a multi-step process that involves understanding the problem, devising a plan, carrying out the plan, and then looking back to review the results. Our photocopy scenario illustrates this process in action. First is comprehension: What are the costs involved with each copying option? Next, comes planning: In this step, the break-even analysis is selected as the method to find the solution.

Implementation follows through algebraic manipulation of the cost equations. The final step, reflection, often overlooked, is crucial for verifying the correctness and reasonableness of the solution. When solving the equation, remember that results should be interpreted contextually—here, fractional copies are not possible, hence rounding up to the nearest whole number is appropriate. Problem-solving skills in mathematics, apart from being essential for academic purposes, are invaluable life skills that help in making effective and logical decisions.