Question

An ice skating rink charges \(\$ 4.75\) for admission and skate rental. If you bring your own skates, the admission is \(\$ 3.25 .\) You can buy a pair of ice skates for \(\$ 45 .\) How many times must you go ice skating to justify buying your own skates?

Short answer

You need to go ice skating 30 times to justify buying your own skates.

Step-by-step solution

Define Costs

Firstly define two types of costs, the cost of ice skating with rented skates \(Cost_{rent}\) and the cost of ice skating with purchased skates \(Cost_{own}\). \(Cost_{rent}\) equals the product of the number of times skating \(n\) and the combined admission and rental fee (\$4.75). \(Cost_{own}\) equals the sum of the cost of the ice skates (\$45) and \(n\), representing the number of times ice skating, multiplied by the admission fee (\$3.25).

Formulate an Equation

The break-even point forms when \(Cost_{rent} = Cost_{own}\). Hence, the equation becomes: \(4.75n = 45 + 3.25n\).

Solve the Equation

You simplify the equation to identify the break-even point by taking the cost of the skate rental (\$4.75) minus the admission (\$3.25) from both sides, which gives: \(1.5n = 45\). Then you divide both sides by 1.5 to find \(n = 30\). Thus, you must go ice skating 30 times to justify the purchase of your own skates.

Key concepts

Algebraic Problem-Solving

Algebraic problem-solving is a systematic approach to find the answers to problems using algebra, a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In real-world applications, such as the ice skating rink example, algebra helps in finding a practical solution to a question - in this case, analyzing costs associated with different options for ice skating.

When faced with a problem, the first step in algebraic problem-solving is to define the variables. In our example, the variable is the number of times you need to skate, represented as n. We then establish two cost scenarios: Cost_rent for renting skates each time and Cost_own for skating after purchasing your own skates.

Understanding how to translate a word problem into an algebraic expression is crucial. This involves identifying the costs or values that change and those that remain constant, setting up an equation based on these values, manipulating the equation to isolate the variable, and finally solving for that variable.

Formulating Equations

To solve real-life problems using algebra, we must first formulate equations that accurately represent the situation. In our scenario, the cost analysis is based on different options of ice skating with either rented or owned skates.

To formulate the correct equation, we consider the fixed cost of purchasing skates (\text{\(45}) and the variable costs of skating each time. With rented skates, the cost is \text{\)4.75} times the number of skates n, and with owned skates, the cost is a fixed \text{\(45} plus \text{\)3.25} times the number of skates n. The equation 4.75n = 45 + 3.25n is thus derived to find the break-even point.

Formulating equations requires an understanding of both the problem at hand and algebraic principles. It is about translating the verbal description into a numerical and variable form, establishing relationships between different elements within a problem, and ensuring that the equation is balanced and reflects the real-world scenario.

Cost Analysis Mathematics

Cost analysis mathematics involves using algebraic techniques to analyze and compare costs in order to make financial decisions. In our ice skating example, cost analysis is used to determine whether it is more economical to rent or buy skates based on the frequency of use.

To conduct a cost analysis, we apply mathematical operations to compare the total costs of two or more options. By setting up an equation where the total costs are equal, we can solve for the break-even point. In this example, after 30 visits, the cost of owning your skates becomes equal to the cost of renting them, making 30 the break-even point.

Understanding the cost analysis in mathematical terms is pivotal for making informed financial decisions. It involves identifying all related costs, both variable and fixed, and how they accumulate over time. It's not just about finding the point where costs are even; it's also important to consider the long-term value and how the initial investment compares to the ongoing expenditure.