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 \( m \):
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Subtract \( 37.50m \) from both sides: \[ 45m - 37.50m + 20 = 65 \] \[ 7.5m + 20 = 65 \]
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Subtract \( 20 \) from both sides: \[ 7.5m = 65 - 20 \] \[ 7.5m = 45 \]
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Divide both sides by \( 7.5 \): \[ m = \frac{45}{7.5} \]
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Calculate \( m \): \[ m = 6 \]
So, the total cost at either gym will be the same after 6 months.