Nick wants to join a gym, and he is deciding between two options. Infinity Fitness charges $45 per month, plus a one-time registration fee of $20. The Power Zone charges $37.50 per month, plus a one-time registration fee of $65.

Question

Which equation can you use to find m, the number of months it would take for the total cost at either gym to be the same?
After how many months would the total cost at either gym be the same?
months

Answer 1

To find the number of months \( m \) at which the total costs for both gyms are the same, we first need to set up the total cost equations for each gym.

For Infinity Fitness:

• Monthly cost: \( 45m \)
• One-time registration fee: \( 20 \)
• Total cost: \( C_1 = 45m + 20 \)

For Power Zone:

• Monthly cost: \( 37.50m \)
• One-time registration fee: \( 65 \)
• Total cost: \( C_2 = 37.50m + 65 \)

To find \( m \) when the costs are equal, we set the equations equal to each other:

\[ 45m + 20 = 37.50m + 65 \]

Now, we can solve for \( m \):

1. Subtract \( 37.50m \) from both sides: \[ 45m - 37.50m + 20 = 65 \] \[ 7.5m + 20 = 65 \]

2. Subtract \( 20 \) from both sides: \[ 7.5m = 65 - 20 \] \[ 7.5m = 45 \]

3. Divide both sides by \( 7.5 \): \[ m = \frac{45}{7.5} \]

4. Calculate \( m \): \[ m = 6 \]

So, the total cost at either gym will be the same after 6 months.