Problem 46 - Algebra 1

Question

A rock-climbing gym charges nonmembers \(\$ 16\) per day to use the gym and \(\$ 8\) per day for equipment rental. Members pay a yearly fee of \(\$ 450\) for unlimited climbing and \(\$ 6\) per day for equipment rental. Write and solve an equation to find how many times you must use the gym to justify becoming a member.

Step-by-Step Solution

Answer

You would need to use the gym for 25 days in the year to justify becoming a member.

Step 1: Define the variables

Let \(n\) represent the number of days the gym will be used in the year.

Step 2: Set up the equation

The yearly cost for a non-member is the daily fee times the number of days (\(16n + 8n\)) and for a member it is the annual fee plus the daily equipment rental fee times the number of days (450 + 6n). Set these two equations equal to each other: \[16n + 8n = 450 + 6n\]

Step 3: Simplify the equation

Combine like terms and isolate \(n\). This yields \[24n = 450 + 6n\] \[18n = 450\]

Step 4: Solve for n

Divide both sides by 18 to solve for \(n\): \[n = 450 / 18 = 25\]

Key Concepts

Membership Costs

When deciding on whether to purchase a gym membership or pay as a non-member, it's crucial to evaluate both options economically. A gym typically offers different pricing structures. For instance, in this scenario, a non-member pays $16 per day for gym usage and $8 for equipment rental, summing up to $24 for each visit. On the other hand, a member enjoys unlimited access with an upfront cost of $450 per year, plus a reduced equipment rental fee of $6 per day.

• Non-member: Cost per day is calculated as the sum of the gym fee and rental, which totals to $24.
• Member: The annual fee is $450, and equipment rental is $6 per day.

Understanding these costs helps in comparing expenses to determine which option provides better value.

Variable Definition

In order to model real-world problems using algebraic equations, we need to define variables accurately. Variables are symbols that represent unknown values or quantities. In this exercise, the variable is essential for determining the break-even point for gym membership.
The variable chosen here is \( n \), which stands for the number of days an individual plans to use the gym in a year. Defining \( n \) helps us set up equations that compare costs.

• Without this definition, it would be impossible to set up an equation that reflects the actual usage scenario.
• Choosing \( n \) allows flexibility in calculations, as it can easily adapt to different usage frequencies.

With a clear understanding of what \( n \) represents, the algebraic model can be accurately constructed and solved.

Equation Solving

Solving equations involves manipulating mathematical statements to find the values of variables that satisfy them. Let’s break down the steps.
The first step is setting up the equation. For this problem, the cost comparison between being a non-member and a member is expressed as:\[ 16n + 8n = 450 + 6n \]

• Combine like terms on the left-hand side: \( 24n \)
• Then, set up the simplified equation: \( 24n = 450 + 6n \)

To solve for \( n \), isolate the variable.
Subtract \( 6n \) from both sides to get:\[ 18n = 450 \]Finally, divide both sides by 18:\[ n = \frac{450}{18} \]Which simplifies to 25. This means using the gym 25 times in a year justifies becoming a member. The process highlights the importance of systematic equation solving to derive meaningful conclusions.