Problem 27 - College Algebra Essentials

Question

You are choosing between two health clubs. Club A offers membership for a fee of \(\$ 40\) plus a monthly fee of \(\$ 25 .\) Club \(B\) offers membership for a fee of \(\$ 15\) plus a monthly fee of \(\$ 30\). After how many months will the total cost at each health club be the same? What will be the total cost for each club?

Step-by-Step Solution

Answer

The total costs for each club will be the same after roughly 1.67 months, and the total cost for each club at that time will be \$56.67.

Step 1: Writing the equations

Write down the equations for both health clubs using the formula: total cost = initial fee + (monthly fee * number of months). Thus, for Club A it would be \( C_A = 40 + 25m \) and for Club B it would be \( C_B = 15 + 30m \), where \( m \) is the number of months and \( C_A, C_B \) are the total costs for Clubs A and B respectively.

Step 2: Setting up the equations

Since the problem asks for the point at which the costs are the same, set these two equations equal to each other to get \( 40 + 25m = 15 + 30m \).

Step 3: Solving the equation

Solve the equation for \( m \) to get the number of months where costs are the same. Following algebraic rules, subtract \( 25m \) from both sides to get \( 15m = 25 \), and then divide both sides by 15 to get \( m = 5/3 \). So after roughly 1.67 months, the total costs will be the same.

Step 4: Calculating total cost

Substitute \( m \) into one of the equations—which will now be equal to obtain the total cost. So the total cost for each club when \( m = 5/3 \) in \( C_A = 40 + 25(5/3) \) which gives a result of \( C_A = C_B = \$56.67 \)